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Fuzzy Linear Codes

Fuzzy Linear Codes FUZZY INFORMATION AND ENGINEERING 2018, VOL. 10, NO. 4, 418–434 https://doi.org/10.1080/16168658.2019.1706959 a a b c S. Atamewoue Tsafack , S. Ndjeya , L. Strüngmann and C. Lele Department of Mathematics in the Faculty of Science, at the University of Yaounde I, Yaounde, Cameroon; Faculty of Computer Sciences Institute of Applied Mathematics, Mannheim University of Applied Sciences, Mannheim, Germany; Department of Mathematics and Computer Sciences in the Faculty of Science at the University of Dschang, Cameroon ABSTRACT KEYWORDS Code over Galois ring; fuzzy In this paper, we define the notion of fuzzy linear code, fuzzy cyclic linear code; fuzzy cyclic code; code over a Galois ringZ k , fuzzy Gray map and we use it to construct fuzzy generalised gray map fuzzy Z -linear codes and fuzzy Z -cyclic codes. k k p p 1. Introduction When we study a subject, we always encode its information and decode the received infor- mation, this is what the classical code theory deals with, and the information which we handle are certain. However, for uncertain information, the classical code theory has less efficient methods. Since fuzzy mathematics has nice applications when dealing with fuzzi- ness, we try to use the methods of fuzzy mathematics to conduct fuzzy information. The notion of fuzzy subsets was first developed by Zadeh [1] in which imprecise knowledge can be used to define an event. The importance of fuzzy sets comes from the fact that it can deal with imprecise and inexact information. The concept of fuzzy modules was intro- duced by Negoita and Ralescu [2] while the notion of fuzzy submodule was introduced by Maschinchi and Zahedi [3]. Shum and De Gang [4] introduced the concept of fuzzy linear code over finite fields. If the data from the information channel is uncertain, then the ordinary method of decoding can not deal with it. For instance, let c be an information of the subject that we study. Since the data from the information channel is uncertain, we can not make sure wether or not the subject (to the arrival) has this information again. We only can estimate the degree of which it possesses the information c and assign a corresponding degree in [0,1]. If for every information there is such a number corresponding to it, then we can get a fuzzy subset A of the block code, we call it the fuzzy code. In this paper, we mainly define the fuzzy linear code and fuzzy cyclic code over the ring Z . We also use the fuzzy generalised Gray map to define Z -fuzzy linear codes and we k k p p study the basic properties of all these types of codes. CONTACT S. Atamewoue Tsafack surdive@yahoo.fr © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 419 2. Preliminaries In this section, we shall formulate the preliminary definitions and results that are required in the sequel (for references see [2,5]). Definition 2.1: Let S be a non-empty set. A fuzzy subset A of S is a function of S into the closed interval [0, 1]. Definition 2.2: Let S be a non-empty set with an additive and multiplicative operation, and let A and B be two fuzzy subsets of S. Then: • (A ∩ B)(x) = min{A(x), B(x)}, for all x ∈ S. • (A ∪ B)(x) = max{A(x), B(x)}, for all x ∈ S. • A = B if and only if A(x) = B(x), for all x ∈ S. • (A + B)(x) = max{A(y) ∧ B(z) | x = y + z with y, z ∈ S}, for all x ∈ S. • (AB)(x) = max{A(y) ∧ B(z) | x = yz with y, z ∈ S}, for all x ∈ S. • A ⊆ B if and only if A(x) ≤ B(x), for all x ∈ S. From now on (R, +, ·) will denote a commutative unitary ring or simply Z k, where p is a prime integer and k ∈ N, k = 0. Definition 2.3: A R-module M consists of an abelian group (M, ⊕) andanoperation ∗ : R × M → M (called scalar multiplication, usually just written by juxtaposition, i.e. as rx instead of r ∗ x for r ∈ R and x ∈ M) such that for all r, s ∈ R, x, y ∈ M,wehave (a) r(x ⊕ y) = rx ⊕ ry, (b) (r + s)x = rx ⊕ sx, (c) (rs)x = r(sx), (d) 1 x = x where 1 is the multiplicative identity of the ring R. R R From [6,7] we recall the following definition in the fuzzy linear space. Definition 2.4: A fuzzy subset X of a R-module M is called a fuzzy submodule of M if for all x, y ∈ M and r ∈ R,wehave. (a) X(x ⊕ y) ≥ min{X(x), X(y)}. (b) X(rx) ≥ X(x). Remark 2.5: If X is a fuzzy submodule of M, then from b) in Definition 2.4 follows (∀ x ∈ M)(X(0) ≥ X(x)). Definition 2.6: Let A be a fuzzy subset of a nonempty set M.For t ∈ [0, 1], the sets A ={x ∈ M : A(x) ≥ t} and A ={x ∈ M : A(x) ≤ t} are called the upper t-level cut and lower t-level cut of A, respectively. Proposition 2.7: Let M be a R-module. A nonempty subset N of M is a submodule of M if and only if the characteristic function of N is a fuzzy submodule. 420 S. ATAMEWOUE TSAFACK ET AL. Proposition 2.8: A is a fuzzy submodule of a R-module M if and only if for all α, β ∈ R, x, y ∈ M, we have A(αx ⊕ βy) ≥ min{A(x), A(y)}. Proof: The proof is similar to the one for fields in [4], just change a field by ring. Definition 2.9: A fuzzy subset I of a ring R is called a fuzzy ideal of R if for each x, y ∈ R: (a) I(x − y) ≥ min{I(x), I(y)}. (b) I(x · y) ≥ max{I(x), I(y)}. Let G be a group and R a ring. We denote by RG the set of all formal linear combinations of the form α = a g (where a ∈ R and a = 0 almost everywhere, that is only a finite g g g g∈G number of coefficients are different from zero in each of these sums). Let α = a g and β = b g in RG. We define their sum in RG componentwise g g g∈G g∈G by: α + β = (a + b )g and their product by: αβ = a b gh. g g g g∈G g,h∈G With the operations above, RG is a unitary ring, with 1 = u g where the coefficient g∈G corresponding to the unit element of the group is equal to 1 and u = 0 for every other element g ∈ G. We can also define a product of elements in RG by elements λ ∈ R as λ( a g) = g∈G (λa )g. g∈G Definition 2.10: The set RG with the operations defined above is called the group ring of G over R,if R is commutative, then RG is called the group algebra of G over R. Definition 2.11: Let A be a fuzzy subset of the group ring (RG) which is the group algebra of x over the ring Z k, note that x is an invertible element of Z k. If for all α, β ∈ RG, p p (a) A(α.β) ≥ max{A(α), A(β)}; (b) A(α − β) ≥ min{A(α), A(β)}, then A is call a fuzzy ideal of RG. Proposition 2.12: A is a fuzzy ideal of RG if and only if for all t ∈ [0, 1], if A =∅, then A is an t t ideal of RG. Proof: The proof follows from the transfer principle in [8]. Definition 2.13: Let n be an integer. A linear code of length n over Z k is a submodule of Z . In contrast to vector spaces, modules do not admit a basis in general. However modules possess a generating family and therefore a generating matrix, but the decomposition of the elements with respect to this family is not necessarily unique. Definition 2.14: A generating matrix of some linear code over Z is a matrix in M(Z ), k k p p where the lines are the minimal generating family of code. FUZZY INFORMATION AND ENGINEERING 421 Definition 2.15: Let C k and C be two linear codes over Z k with generating matrices G p p and G respectively. The codes C k and C are equivalent if there exists a permutation matrix P,suchthat G = GP (the product of the two matrices G and P). Definition 2.16: Let C be a linear code of length n overZ ,the dual of the code C which k k k p p p ⊥ n ⊥ we denote by C is the submodule of Z defined by: C ={a| for all b ∈ C , a, b = 0}, k k k p p p p where ,  is the inner product. Remark 2.17: C is also a linear code over Z k. Definition 2.18: A linear code C k of length n over Z k is cyclic if it is invariant by the shift p p map s, define by s((a , ... , a )) = (a , a , ... , a ). i.e. if (a , ... , a ) ∈ C , then 0 n−1 n−1 0 n−2 0 n−1 s((a , ... , a )) ∈ C . 0 n−1 3. Fuzzy Linear Code Over Z In this section are defined and studied fuzzy linear linear code over the Galois ringZ . There are also characterised by using the transfer principle [8]. Definition 3.1: Let M = Z be a Z -module. A fuzzy submodule A of M is called a fuzzy k p linear code of length n over Z k. Example 3.2: Let Z be a ring, then Z is a Z -module. 4 4 4 Let A : Z → [0, 1] be the map such that A(0) = A(1) = A(2) = A(3) = t (t ∈ [0, 1]), then A is a fuzzy subset ofZ .Itisobviousthat A is a fuzzyZ -module. Therefore A is a fuzzy linear 4 4 code over Z . Proposition 3.3: Let A be a fuzzy subset of Z . A is a fuzzy linear code of length n over Z if and only if for any t ∈ [0, 1], if A =∅, then A t t is a linear code of length n over Z . Proof: Use the transfer principle in [8]. Corollary 3.4: Let C be a subset of Z . C is a linear code of length n overZ if and only if the characteristic function χ of C is a fuzzy linear code over Z . Proposition 3.5: Let A be a fuzzy subset of Z . A is a fuzzy linear code of length n over Z k if and only if the characteristic function of any upper t-level cut A =∅ for t ∈ [0, 1] is a fuzzy linear code of length n over Z k. Proof: Uses Proposition 3.3 and Corollary 3.4. Remark 3.6: We remark the following: (i) In the Example 3.2, A = Z (for all t ∈ [0, 1]) which is a linear code over Z . t 4 4 422 S. ATAMEWOUE TSAFACK ET AL. (ii) If M is a module over the ring Z and A a fuzzy linear code A on M such that ∀ x ∈ M, A(x) = t (where t ∈ [0, 1]), then A is called the trivial fuzzy linear code over Z . Example 3.7: Consider a fuzzy subset A of Z as follows: 1if x = 0; ⎪ 1 if x = 1; A : Z → [0, 1], x → ⎪ if x = 2; if x = 3. Then A is a fuzzy submodule of the Z -module Z , hence A is a fuzzy linear code over Z . 4 4 4 Remark 3.8: Let A be a fuzzy linear code of length n over Z k.Since Z is a finite set, the image Im(A) ={A(x) | x ∈ Z } is finite as well. Assume that all elements in Im(A) satisfy: t > t > ··· > t (where t ∈ [0, 1]) i.e. Im(A) has m elements. Since A is a linear code over 1 2 m i t Z k,let G be its generator matrix. Thus A can be determined by m matrixes G , G , ... , G t t t t p i 1 2 m (see Theorem 4.7 ). Definition 3.9: Let S be a non-empty set. Let x : S → [0, 1] be a fuzzy subset of S, (where x ∈ S, t ∈ [0, 1]) defined by: t if x = y; x (y) = , for all y ∈ S. 0if x = y. x is called a fuzzy singleton or fuzzy point of S. Definition 3.10: Let A and A be two fuzzy linear codes over Z of thesamelength n. 1 2 The residual quotient of A and A denoted by (A : A ) is the fuzzy subset of Z defined 1 2 1 2 by: (A : A )(r) = sup{t ∈ [0, 1] | r A ⊆ A } for all r ∈ Z . 1 2 t 2 1 That is (A : A ) ={r | r A ⊆ A , r is fuzzy singleton of Z }. 1 2 t t 2 1 t Theorem 3.11 ([2]): Let A and A be two fuzzy linear codes of length n over Z , then the 1 2 residual quotient (A : A ) of A and A is a fuzzy ideal of Z . 1 2 1 2 Definition 3.12: Let A and B be two fuzzy submodules of a module M = Z over the ring Z . We say that A is orthogonal to B if Im(B) ={1 − c | c ∈ Im(A)} and for all t ∈ [0, 1], B = 1−t (A ) ={y ∈ M | x, y = 0, for all x ∈ A }, where ,  is the inner product on M t t Let A and B be two fuzzy submodules of a module M. We denote A orthogonal to B by A ⊥ B. FUZZY INFORMATION AND ENGINEERING 423 Example 3.13: Consider the fuzzy submodules A and B of Z defined as follows: ⎧ ⎧ 1 3 ⎪ ⎪ ⎪ ⎪ if x = 0; if x = 0; ⎪ ⎪ ⎪ ⎪ 2 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ ⎨ if x = 1; ⎨ if x = 1; 4 2 A : Z → [0, 1], x → and B : Z → [0, 1], x → 4 4 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ if x = 2; ⎪ if x = 2; ⎪ ⎪ ⎪ ⎪ 3 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎩ ⎩ if x = 3. if x = 3. 4 2 We have: A ={0} and B = Z , 1/2 1/2 4 A = Z and B ={0}, 1/4 4 3/4 A ={0, 2} and B ={0, 2}. 1/3 2/3 Therefore A ⊥ B. Remark 3.14: Let A be a fuzzy submodule of a module M such that for all x ∈ M, A(x) = γ (with γ ∈ [0, 1]), then it does not exist a fuzzy subset B of M such that A ⊥ B. The Remark 3.14 shows that the orthogonal of some fuzzy submodule does not always exist, so it is important to see under which conditions the orthogonal of a fuzzy submod- ule exists. The following theorem shows the existence of the orthogonal of some fuzzy submodule. Theorem 3.15: Let A be a fuzzy submodule of a finite module M = Z . Then there exists a fuzzy submodule B of M such that A ⊥ B if and only if |Im(A)| > 1 and for any γ ∈ Im(A) there exist  ∈ Im(A) such that A = (A ) . Proof: Let A be a fuzzy submodule of M. Assume that |Im(A)|= m > 1 and for any γ ∈ Im(A) there exists  ∈ Im(A) such that A = (A ) . Assume that Im(A) ={t > t > ··· > t }. Define the sets M ={x ∈ M | A(x) = t },for 1 2 m i i i = 1, ... , m. These sets form a partition of M. We define a fuzzy set B as follows: B : M → [0, 1], x → 1 − t ,if x ∈ M . m−i+1 i Since Im(A) ={t > t > ··· > t },wehave A ⊆ A ⊆ ··· ⊆ A .Asforany γ ∈ Im(A) 1 2 m t t t 1 2 m there exists  ∈ Im(A) such that A = (A ) , with the properties that we know about the orthogonal over the finite module, we conclude that A = (A ) .Thus B = t t 1−t i m−i+1 m−i+1 {x ∈ M | B(x) ≥ 1 − t }= M ∪ M ∪ ··· ∪ M = A = (A ) .Thus B is the fuzzy m−i+1 i i−1 1 t t i m−i+1 submodule we need. Conversely if there exists a fuzzy submodule B of M such that A ⊥ B, then by Definition 3.12, |Im(A)| > 1. Since ∀ t ∈ [0, 1], B = (A ) , then ∀γ ∈ Im(A), there exist 1−t t ∈ Im(A),suchthat A = (A ) ,because Im(B) ={1 − t | t ∈ Im(A)}. The following result shows the uniqueness of the orthogonal of some fuzzy submodule. Theorem 3.16: Let A, B and C be three fuzzy submodules of a module M, such that A ⊥ Band A ⊥ C, then B = C. 424 S. ATAMEWOUE TSAFACK ET AL. Proof: Assume that A ⊥ B and B ⊥ C.Let t ∈ [0, 1], and y ∈ B . Then x, y = 0, for all x ∈ 1−t A .Thus y ∈ C and B ⊆ C . Therefore C ⊆ B . In the same way, we show that B ⊆ C . t 1−t 1−t 1−t t t t t Therefore B = C. Corollary 3.17: Let A be a fuzzy submodule of a finite module M such that there exists a fuzzy set B on M orthogonal to A, then B is a fuzzy submodule of M. ⊥ ⊥ ⊥ Corollary 3.18: Let A be a fuzzy submodule of M. If A exists, then (A ) = A. Definition 3.19: Let A and B be two fuzzy linear codes overZ . A and B are equivalent fuzzy linear codes over Z if for all t ∈ [0, 1], the linear codes A and B are equivalent. t t Example 3.20: (1) All fuzzy linear code are equivalent to itself. (2) Let C and C be two equivalent linear codes of length n over Z . We define two G G k 1 2 p equivalent fuzzy linear codes as follows: 1if x ∈ C ; A : Z → [0, 1], x → 0 otherwise. and 1if x ∈ C ; B : Z → [0, 1], x → 0 otherwise. n n Thus A = C and B = C , A = Z and B = Z . 1 G 1 G 0 0 1 2 k k p p Remark 3.21: Let A and B be two equivalent fuzzy linear codes over Z , then Im(A) = Im(B). Let’s draw the communication channel as follows: k 2 n 3 3 Assume that F = Z and F = Z , that means that k = 2and n = 3. Let C ⊆ F be a 2 2 linear code over F, in the classical case, when we send a codeword c = (101) ∈ C through a communication channel, the signal receive can be read as c= (0.98, 0.03, 0.49) and mod- ulate to c = (100). Thus to know if c belong to the code C, we use syndrome calculation [9]. Since the modulation have gave a wrong word, we can consider that c have more infor- mation than c , in the sense that we can estimate a level to which a word 0 is modulate to 1, and a word 1 is modulate to 0. Therefore it is possible to use the idea of fuzzy logic to recover the transmit codeword. Let a linear code C ⊆ Z .Toeach c ∈ C, we find t ∈ [0, 1] such that t estimate the degree of which the element of R ,obtainfrom c through the transmission channel belong to the 3 3 code C.ThusinZ the information that we handle are certain, whereas inR there are uncer- tain. When we associate to all elements of Z the degree of which its correspond element obtain through the transmission channel belong to Z , then we obtain a fuzzy code. If the fuzzy code are fuzzy linear code, then we can recover the code C just by using the upper t-level cut. Thus we deal directly with the uncertain information to obtain the code C. FUZZY INFORMATION AND ENGINEERING 425 The following example illustrate this reconstruction of the code by using uncertain information in the case of fuzzy linear code. Example 3.22: LetZ ={000, 001, 010, 100, 110, 101, 011, 111} and C ={000, 001, 110, 111} be a linear code over Z . Assume that after the transmission we obtain respectively {000; 0.01, 01; 1.01, 10; 1.001, 1, 0.999}.Let ⎪ {1} if x = 000; {0.99} if x = 001; {0.9} if x = 010; {0.9} if x = 100; A : Z → [0, 1] such that x → {0.99} if x = 110; ⎪ {0.9} if x = 101; ⎪ {0.9} if x = 011; {0.99} if x = 111. Then by finding a t ∈ [0, 1] such that A ={x ∈ Z | A(x) ≥ t}= C,weobtain t > 0.9. Thus, for t = 0.99, we are sure that the receive codeword is in C. 4. Fuzzy Cyclic Codes Over Z k In this section, we will study the case where the integers n and p are coprime. Definition 4.1: A fuzzy submodule A of the module Z is called a fuzzy cyclic code of length n over Z k if for all (a , a , ... , a ) ∈ Z ,wehave A((a , a , ... , a )) ≥ 0 1 n−1 n−1 0 n−2 A((a , a , ... , a )). 0 1 n−1 Proposition 4.2: Let A be a fuzzy submodule A of the module Z . A is a fuzzy cyclic code on n n Z if and only if for all t ∈ [0, 1], if A =∅, then A is a cyclic code on Z . t t k k p p Proof: The proof uses the transfer principle from [8]. As well as the Proposition 3.5 in the linear case, we have the following result in the cyclic case. Proposition 4.3: Let A be a fuzzy module A on the module Z . A is a fuzzy cyclic code on Z if and only if the characteristic of any upper t-level cut A =∅ for t ∈ [0, 1] is a fuzzy cyclic code on Z . n n Proposition 4.4: A is a fuzzy cyclic code on Z if and only if for all (a , a , ... , a ) ∈ Z , 0 1 n−1 k k p p then A((a , a , ... , a )) = A((a , a , ... , a )) = ··· = A((a , a , ... , a , a )). 0 1 n−1 n−1 0 n−2 1 2 n−1 0 426 S. ATAMEWOUE TSAFACK ET AL. Proof: Assume that A is a fuzzy cyclic code on Z . Then A((a , a , ... , a )) ≤ A((a , a , ... , a )) ≤ ··· ≤ A((a , a , ... , a , a )) 0 1 n−1 n−1 0 n−2 1 2 n−1 0 ≤ A((a , a , ... , a )). 0 1 n−1 Therefore A((a , a , ... , a )) = A((a , a , ... , a )) = ··· = A((a , a , ... , a , a )). 0 1 n−1 n−1 0 n−2 1 2 n−1 0 The converse is straightforward by Definition 4.1. Proposition 4.5: A is a fuzzy cyclic code of length n over Z if and only if for all t ∈ [0, 1], if A =∅, then A is an ideal of the factor ring Z [X]/(X − 1). t t n n Proof: Let φ be a mapping defined in [5] as follows, φ : Z → Z [X]/(X − 1),suchthat k p n−1 c = (c , ... , c ) → φ(c) = c X . φ is an isomorphism of Z -module, which sends a 0 n−1 i k i=0 p cyclic code over Z onto the ideals of the factor ring Z [X]/(X − 1). k k p p Let A be a fuzzy subset of Z . Assume that A is a fuzzy cyclic code over Z k. Let t ∈ [0, 1] such that A =∅, then A is a cyclic code over Z . Therefore, ∀ t ∈ [0, 1], A t t t is an ideal of Z [X]/(X − 1). Conversely, assume that for all t ∈ [0, 1] such that A =∅, A is an ideal of factor ring t t n n Z [X]/(X − 1).Since A is an ideal of factor ring Z [X]/(X − 1), then A is a submodule k k t t p p of Z -module Z . Hence A =∅, is a linear code over Z , then A is a fuzzy linear code. k k p k p Such as φ is define, A is a cyclic code over Z , for all t ∈ [0, 1]. Hence A is a fuzzy cyclic code over Z . Proposition 4.6: A is a cyclic fuzzy code of length n if and only if A is a fuzzy ideal of the group algebra RG, which is the group algebra of x over the finite ring Z . Proof: Let A be a fuzzy cyclic code. For any α, β ∈ RG, we have A(α − β) ≥ min{A(α), A(β)} since A is a fuzzy Z k-module on Z . 2 n−1 If α ∈ RG, then A(xα) ≥ A(α), A(x α) ≥ A(α), ··· , A(x α) ≥ A(α).Sofor β ∈ RG, n−1 i n−1 with β = l x ,(l ∈ Z k) we conclude A(α.β) = A(l α + l xα + ··· + l x α) ≥ i i 0 1 n−1 i=0 n−1 min{A(l α), ... , A(l x α)}≥ A(α). Similarly we can also show that A(α.β) ≥ A(β). 0 n−1 Hence A(α.β) ≥ max{A(α), A(β)}. Conversely, assume that A is a fuzzy ideal of the group algebra RG. (Note that RG is a module that has G = x as base). Since A is a fuzzy ideal of RG, also A is a fuzzy submodule of RG. n n−1 For any (a , ... , a ) ∈ Z we associate α = (a + a x + ··· + a x ). Then 0 n−1 0 1 n−1 n−1 n−1 A(a + a x + ··· + a x ) = A(x(a + a x + ··· + a x )) ≥ max{A(x), A(a + n−1 0 n−2 0 1 n−1 0 n−1 n−1 a x + ··· + a x )}≥ A(a + a x + ··· + a x ). Therefore A((a , a , ... , a )) 1 n−1 0 1 n−1 n−1 0 n−2 ≥ A((a , ... , a )). Hence A is a fuzzy cyclic code. 0 n−1 Since Z k is a finite ring, also Im(A) ={A(x) ∈ [0, 1]|x ∈ Z } is finite. Let Im(A) ={t > (k) t > ··· > t }, then A ⊆ A ⊆ ··· ⊆ A ⊆ A = Z .Let g (X) ∈ Z [X] be the gen- 2 m t t t t 1 2 m−1 m k p (k) erator polynomial of A . Note that g (X) is the Hensel lift of order k of some polynomial i i FUZZY INFORMATION AND ENGINEERING 427 (k) n n g (X) ∈ Z [X] which divides X − 1. The cyclic code g (X)⊂ Z k[X]/(X − 1) is called i p the lift code of the cyclic code g (X)⊂ Z [X]/(X − 1). For more information about Hensel i p lifting see [5]. (k) (k) Since A ⊆ A ⊆ ··· ⊆ A ⊆ A = Z , it follows that g (X)|g (X), i = 1, ... , t t t t 1 2 m−1 m k i+1 i m − 1. (k) (k) We can define the polynomial h (X) = (X − 1)/g (X) whichiscalledthe check i i (k) polynomial of the cyclic code A =g (X), i = 1, ... , m. i i (k) (k) (k) Theorem 4.7: Let G ={g (X), g (X), ... , g (X)} be a set of polynomials in Z [X], 1 2 p (k) (k) such that g (X) divide X − 1, i = 1, ... , m. If g (X) | g (X) for i = 1, 2, ... , m − 1 and i+1 i (k) (k) g (X)= Z , then the setG determines a fuzzy cyclic code A and {g (X)| i = 1, ... , m} is the family of upper level cut cyclic subcodes of A. (k) (k) (k) (k) Proof: Since g (X) | g (X),wehave g (X)⊆g (X) for i = 1, 2, ... , m − 1. Choose i+1 i i i+1 t ∈ [0, 1] such that t > t > ··· > t . i 1 2 m (k) Let A =g (X) for i = 1, 2, ... , m − 1. We define A as follows. i i (k) t ,if c ∈g (X); A(c) = (k) (k) t,if c ∈g (X)\g (X), i = 2, ... , m. i i−1 n−1 n n i where φ : Z → Z k[X]/(X − 1), c = (c , ... , c ) → φ(c) = c X is an isomor- 0 n−1 i p i=0 phism of Z -module. (k) Since for all t ∈ [0, 1], A =g (X) is a cyclic code as it is an ideal of the principal ring i t i i (k) Z [X]/(X − 1), i = 1, ... , m,also A is a fuzzy cyclic code and {g (X)| i = 1, ... , m} is the family of upper level cut cyclic subcodes of A. (k) Corollary 4.8: With the same notations and hypothesis as in Theorem 4.7, if g (X) = Z , m k (k) then the setG determines a fuzzy cyclic code A and {g (X)| i = 1, ... , m}∪ Z k[X]/(X − 1) i p is the family of upper level cut cyclic subcodes of A. Proof: Take (k) t ,if c ∈g (X); ⎪ 1 (k) (k) t,if c ∈g (X)\g (X), i = 2, ... , m; i i−1 A(c) = Z [X] (k) 0, if c ∈ \g (X). (X − 1) Proposition 4.9: Let A be the fuzzy cyclic code of length n overZ that can be determined by (k) (k) the set of polynomialG ={g (X) | i = 1, ... , m} as in Theorem 4.7.Ifforallg (X) ∈ G there i i (k) (k) (k) (k) exists g (X) ∈ G such that g (X).g (X) = X − 1, then the set of polynomials {h (X) = j i j i (k) (X − 1)/g (X) | i = 1, ... , m} determines the orthogonal of A. i 428 S. ATAMEWOUE TSAFACK ET AL. Proof: Under the conditions of Theorem 4.7 we define A as follows. (l) 1 − t ,if c ∈h (X); 1 m A (c) = (l) (l) 1 − t,if c ∈h (X)\h (X), i = 2, ... , m. i−1 i ⊥ ⊥ Since the upper level set (A ) is a linear code, A is the orthogonal of A. Theorem 4.10: Let A and A be two fuzzy cyclic codes on Z , then: 1 2 (i) A ∩ A is a fuzzy cyclic code, 1 2 (ii) A + A is a fuzzy cyclic code, 1 2 (iii) A A is a fuzzy cyclic code. 1 2 Proof: Let A and A be two fuzzy cyclic codes of the Z -module Z . 1 2 k (i) Let (a , a , ... , a ) ∈ Z . 0 1 n−1 A ∩ A ((a , a , ... , a )) 1 2 n−1 0 n−2 = min{A ((a , a , ... , a ), A ((a , a , ... , a ))} 1 n−1 0 n−2 2 n−1 0 n−2 ≥ min{A (a , a , ... , a ), A ((a , a , ... , a ))} 1 0 1 n−1 2 0 1 n−1 = A ((a , a , ... , a )) ∩ A ((a , a , ... , a )). 1 0 1 n−1 2 0 1 n−1 Since the intersection of two fuzzy modules is a fuzzy module, we obtain that A ∩ A 1 2 is a fuzzy cyclic code over Z k. (ii) For all (a , ... , a ) ∈ Z ,wehave: 0 n−1 (A + A )((a , a , ... , a )) 1 2 n−1 0 n−2 = max{A ((b , b , ... , b )) ∧ A ((c , c , ... , c )) | b 1 n−1 0 n−2 2 n−1 0 n−2 i + c = a , i = 0, ... , n − 1} i i ≥ max{A ((b , b , ... , b )) ∧ A ((c , c , ... , c ))|b + c = a , i = 0, ... , n − 1} 1 0 1 n−1 2 0 1 n−2 i i i = (A + A )((a , a , ... , a )). 1 2 0 1 n−1 Since A + A is a fuzzy module, we conclude as above that A + A is a fuzzy cyclic 1 2 1 2 code. (iii) It is similar to A + A . 1 2 5. Fuzzy Z -linear Codes After having studied the notion of fuzzy linear code over the ringZ k in the previous section, we are now going to construct fuzzy Z k-linear codes explicitly. 5.1. Fuzzy Gray Map Initially, the code of Gray is an order on the binary sequences of a fixed length n, permitting to enumerate all these sequences while modifying only one bit in order to pass from one FUZZY INFORMATION AND ENGINEERING 429 sequence to the next one. The case that is going to interest us directly is the one of the sequence of length two, for which one has the following Gray code: 0 → 00 1 → 01 2 → 11 3 → 10. Let ψ : Z 2 → Z the Gray map. We are going to define the fuzzy Gray map between two 2 2 fuzzy spaces by the extension principle [10]. 2 2 Definition 5.1: Let ψ : Z → Z be the Gray map, and let F (Z ), F (Z ) be the set of 2 2 2 2 2 2 all fuzzy subsets of Z and Z respectively. The fuzzy Gray map is the map ψ : F (Z ) → 2 2 2 2 F (Z ), such that for any A ∈ F (Z ), ψ(A)(y) = sup{A(x) | y = ψ(x)}. 2 2 Example 5.2: Let 1if x = 0; ⎪ 1 if x = 1; A : Z → [0, 1], x → if x = 2; if x = 3. with the Gray map ψ(0) = 00, ψ(1) = 01, ψ(2) = 11, ψ(3) = 10. 1 1 By the fuzzy Gray map we have ψ(A)(00) = 1, ψ(A)(01) = , ψ(A)(11) = , ψ(A)(10) = 3 3 , hence ψ(A) is a fuzzy linear code. Theorem 5.3: The fuzzy Gray map ψ is a bijection. Proof: This follows from the fact that ψ is a one to one function. As in crisp case, we have the following proposition. Proposition 5.4: If A is a fuzzy linear code over Z 2 and ψ the Gray map, then ψ(A) is not always a fuzzy linear code over the field Z The Gray map allows to construct nonlinear codes as binary images of the linear codes, that is the case of Kerdock, Preparata, and Goethals codes. For a good understanding, we suggest the reader to examine [11,12]. In fact if C is a linear code of length n over Z , then C = ψ(C) is a nonlinear code of length 2n over Z in general [11]. In that way we construct a fuzzy Kerdock code in the following example. 430 S. ATAMEWOUE TSAFACK ET AL. Example 5.5: Let C be a linear code of length 8 over Z with the generator matrix ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ G = , ⎝ ⎠ then its image under the Gray map ψ gives a Kerdock code C (see construction in [9]). Let 1, if x ∈ C; A : Z → [0, 1], x → 0, otherwise. Then A is a fuzzy linear code over Z . Since ψ is a bijection, we construct 1, y ∈ E; ψ(A) : Z → [0, 1], y → 0, otherwise, where E ={y ∈ Z | y = ψ(x) and x ∈ C} Since E is not a linear code over Z ,weconcludethat ψ(A) is a fuzzy Z -linear code but 2 2 not a fuzzy linear code Z . Consequently, ψ(A) is a fuzzy Kerdock code of length 16. Remark 5.6: A fuzzy Z -linear code is not in general a fuzzy linear code over Z 4 2 If we define the fuzzy binary relation R on Z 2 × Z by 1, if y = ψ(x); R (x, y) = 0, otherwise. It is easy to see [13]that ψ(A)(y) = sup{A(x) | y = ψ(x)} can be represented by ψ(A)(y) = sup{min{A(x), R (x, y)}| x ∈ Z }. k−1 As in [5], let : Z → Z be the generalised Gray map. k−1 Definition 5.7: We call the map : F (Z k ) → F (Z ), such that for any A ∈ F (Z k ), p p sup{A(x) | y = (x)},ifasuch x exists; (A)(y) = 0, otherwise. The fuzzy generalised gray map. k−1 Since : Z → Z cannot give more than one image for one element, then Definition 5.7 can be simply write A(x),if y = (x); (A)(y) = 0, otherwise. k−1 k−1 p p Remark 5.8: Let B ∈ F (Z ) such that B(y) = t = 0 for any y ∈ Z . There does not p p exist a fuzzy subset A ∈ F (Z k ) such that (A) = B.Thus is not a bijection map. p FUZZY INFORMATION AND ENGINEERING 431 5.2. Fuzzy Z k-linear Codes k−1 n.p In the following, we will denote by the map from F (Z ) onto F (Z ) which spreads k p the fuzzy generalised Gray map. Definition 5.9: A fuzzy code A over Z is a fuzzy Z -linear code if it is an image under the p k fuzzy generalised Gray map of a fuzzy linear code over the ring Z . Definition 5.10: A fuzzy code A is a fuzzy Z k-cyclic code if it is a fuzzy Z k-linear code and p p if it is the image under the generalised Gray map of a cyclic code over the ring Z . Remark 5.11: A fuzzy Z -linear code is a fuzzy code over the field Z . Example 5.12: (1) Let 1, if e = f = 0; B : Z → [0, 1], w = (a, b, c, d, e, f ) → 0, otherwise. B is a fuzzy linear code of length 6 over Z .Let 1, if z = 0; A : Z → [0, 1], v = (x, y, z) → 0, otherwise. A is a fuzzy linear code of length 3 over Z . Moreover, if B = ψ(A), then B is a fuzzy Z -linear code. (2) Let B : Z → [0, 1], ,if d = e = f = g = h = i = 0and abc ∈ {000, 012, 021, 111, 120, 102, 222, 201, 210}; ⎨ 1 ,if g = h = i = 0, abc ∈{000, 012, 021, 111, v = (a, b, c, d, e, f, g, h, i) → 120, 102, 222, 201, 210} and def ∈{012, 021, 111, 120, 102, 222, 201, 210}; 0, otherwise. B is a fuzzy linear code over Z and B is a fuzzy Z 2-linear code. It is easy to show the next proposition, whose Example 5.5 is a perfect illustration of it. Proposition 5.13: Let B be a fuzzy Z -linear code, then B is not always a fuzzy linear code over the field Z . Since the fuzzy generalised Gray map image of fuzzy linear code is a fuzzy codes over the field Z , we can also construct fuzzy Z -linear codes using the following diagram: p 432 S. ATAMEWOUE TSAFACK ET AL. Example 5.14: (1) Let A : Z → [0, 1] be a linear code such that A has three upper t- level cuts A ⊆ A ⊆ A .Let A = (A ), A = (A ) and A = (A ),wehave A = t t t t t t 3 2 1 t 3 t 2 t 1 t 3 2 1 3 (A ) ⊆ A = (A ) ⊆ A = (A ). We construct A = (A) as follow. t t t 3 t 2 t 1 2 1 ⎪ t ,if y ∈ A ; t ,if y ∈ A ; k−1 n.p 2 A : Z → [0, 1], y → t ,if y ∈ A ; ⎪ t 0, otherwise. (2) Let if x = 0; A : Z → [0, 1], x → if x = 2; if x = 1, 3, be a fuzzy linear code over Z . Then A ={0}, A ={0, 2} and A = Z . 4 1/2 1/3 1/4 4 We construct A ={00}, A ={00, 11} and A = Z , the Gray map image of A , 1/2 1/2 1/3 1/4 2 A and A respectively, we define 1/3 1/4 ⎪ if x ∈ A , y = ψ(x); 1/2 A : Z → [0, 1], y → if x ∈ A \ A , y = ψ(x); 1/3 1/2 if x ∈ A \ A , y = ψ(x). 1/4 1/3 Remark 5.15: A and ψ(A) are the same codes. Proposition 5.16: If for all t ∈ [0, 1], A = (A )(when A = 0) is a linear code over Z , then t t p these two constructions of fuzzy Z -linear codes above give the equivalent fuzzy codes. Proof: This follows directly from the definition of the fuzzy generalised Gray map and the fact that the image under the generalised Gray map of a linear code is not a linear code in general.  FUZZY INFORMATION AND ENGINEERING 433 6. Conclusion In this paper where we study fuzzy coding, we define fuzzy linear codes over the finite ring Z , fuzzy generalised Gray map and fuzzy Z -linear codes. We also investigate some of k k p p their properties and remark that many of them are similar to the classical form. The codes of Kerdock are permit us to show that fuzzy Z -linear codes is not a fuzzy linear code. This work allows us to reinforce the hypothesis of Von Kaenel [14], that the theory of fuzzy sets is a natural setting for this study. Acknowledgments We are grateful to the referee for their valuable suggestions, which have improved this paper. Disclosure statement No potential conflict of interest was reported by the authors. Notes on contributors S. Atamewoue Tsafack is a member of Research Team in Algebra and Logic (ERAL) in the Department of Mathematics in the Faculty of Science at the University of Yaounde I, Cameroon. I works on General Algebra, Fuzzy Logic and Coding Theory. S. Ndjeya is a Senior lecturer in the Department of Mathematics in the Faculty of Science at the University of Yaounde I, Cameroon. He works on Pure and Applied Algebra, and Coding Theory. L. Strüngmann is full Professor in the Faculty of Computer Science Institute of Applied Mathematics at the Mannheim University of Applied Sciences, in Germany. He works on Coding Theory, General Algebra and Bioscience. C. Lele is Full Professor in the Department of Mathematics and Computer Sciences in the Faculty of Science at the University of Dschang, Cameroon. He works on General algebra and Fuzzy logic. References [1] Zadeh LA. Fuzzy sets. Inf Control. 1965;8:338–353. [2] Negoita CV, Ralescu DA. Applications of fuzzy sets and system analysis. Basel and Stuttgart: Birkhäuser verlag; 1975. 191PP. [3] Maschinchi M, Zahedi MM. On L-fuzzy primary submodules. Fuzzy Sets Syst. 1992;49:231–236. [4] Shum KP, De Gang C. Some note on the theory of fuzzy code. BUSEFAL (Bulletin Pour Les Sous Ensembles Flous et Leurs Applications) University of Savoie, France. 2000;81:132–136. [5] Galand F. Construction de codes Z k -linéaires de bonne distance minimale et schémas de dissimulation fondés sur les codes de recouvrement [PhD thesis]. Université de Caen; 2004. [6] Biswas R. Fuzzy fields and fuzzy linear spaces redefined. Fuzzy Sets Syst. 1989;33:257–259. [7] Nanda S. Fuzzy fields and fuzzy linear space. Fuzzy Sets Syst. 1986;19:89–94. [8] Kondo M, Dubek WA. On transfer principle in fuzzy theory. Mathware Soft Comput. 2005;12:41–55. [9] Carlet C. Z -linear codes. IEEE Trans Inf Theory. 1998;44:1543–1547. [10] Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning I, II, III. Inf Sci. 1975;8–9:199–249, 301–357, 43–80. [11] Hammons R, Kumar PV, Calderbank AR, et al. Kerdock, Preparata, Goethals and other codes are linear over Z . IEEE Trans Inf Theory. 1994;40:301–319. [12] Kerdock AM. A class of low-rate nonlinear binary codes. Inf Control. 1972;20:182–187. 434 S. ATAMEWOUE TSAFACK ET AL. [13] Perfilieva I. Fuzzy Function: Theoretical and Practical Point of View. Proceedings of the 7th con- ference of the European Society for Fuzzy Logic and Technology, Aix-les-Bains, France: Atlantis Press; 2011. pp. 480–486. Available from: https://doi.org/10.2991/eusflat.2011.142. [14] Von Kaenel PA. Fuzzy codes and distance properties. Fuzzy Sets Syst. 1982;8:199–204. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Fuzzy Linear Codes

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FUZZY INFORMATION AND ENGINEERING 2018, VOL. 10, NO. 4, 418–434 https://doi.org/10.1080/16168658.2019.1706959 a a b c S. Atamewoue Tsafack , S. Ndjeya , L. Strüngmann and C. Lele Department of Mathematics in the Faculty of Science, at the University of Yaounde I, Yaounde, Cameroon; Faculty of Computer Sciences Institute of Applied Mathematics, Mannheim University of Applied Sciences, Mannheim, Germany; Department of Mathematics and Computer Sciences in the Faculty of Science at the University of Dschang, Cameroon ABSTRACT KEYWORDS Code over Galois ring; fuzzy In this paper, we define the notion of fuzzy linear code, fuzzy cyclic linear code; fuzzy cyclic code; code over a Galois ringZ k , fuzzy Gray map and we use it to construct fuzzy generalised gray map fuzzy Z -linear codes and fuzzy Z -cyclic codes. k k p p 1. Introduction When we study a subject, we always encode its information and decode the received infor- mation, this is what the classical code theory deals with, and the information which we handle are certain. However, for uncertain information, the classical code theory has less efficient methods. Since fuzzy mathematics has nice applications when dealing with fuzzi- ness, we try to use the methods of fuzzy mathematics to conduct fuzzy information. The notion of fuzzy subsets was first developed by Zadeh [1] in which imprecise knowledge can be used to define an event. The importance of fuzzy sets comes from the fact that it can deal with imprecise and inexact information. The concept of fuzzy modules was intro- duced by Negoita and Ralescu [2] while the notion of fuzzy submodule was introduced by Maschinchi and Zahedi [3]. Shum and De Gang [4] introduced the concept of fuzzy linear code over finite fields. If the data from the information channel is uncertain, then the ordinary method of decoding can not deal with it. For instance, let c be an information of the subject that we study. Since the data from the information channel is uncertain, we can not make sure wether or not the subject (to the arrival) has this information again. We only can estimate the degree of which it possesses the information c and assign a corresponding degree in [0,1]. If for every information there is such a number corresponding to it, then we can get a fuzzy subset A of the block code, we call it the fuzzy code. In this paper, we mainly define the fuzzy linear code and fuzzy cyclic code over the ring Z . We also use the fuzzy generalised Gray map to define Z -fuzzy linear codes and we k k p p study the basic properties of all these types of codes. CONTACT S. Atamewoue Tsafack surdive@yahoo.fr © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 419 2. Preliminaries In this section, we shall formulate the preliminary definitions and results that are required in the sequel (for references see [2,5]). Definition 2.1: Let S be a non-empty set. A fuzzy subset A of S is a function of S into the closed interval [0, 1]. Definition 2.2: Let S be a non-empty set with an additive and multiplicative operation, and let A and B be two fuzzy subsets of S. Then: • (A ∩ B)(x) = min{A(x), B(x)}, for all x ∈ S. • (A ∪ B)(x) = max{A(x), B(x)}, for all x ∈ S. • A = B if and only if A(x) = B(x), for all x ∈ S. • (A + B)(x) = max{A(y) ∧ B(z) | x = y + z with y, z ∈ S}, for all x ∈ S. • (AB)(x) = max{A(y) ∧ B(z) | x = yz with y, z ∈ S}, for all x ∈ S. • A ⊆ B if and only if A(x) ≤ B(x), for all x ∈ S. From now on (R, +, ·) will denote a commutative unitary ring or simply Z k, where p is a prime integer and k ∈ N, k = 0. Definition 2.3: A R-module M consists of an abelian group (M, ⊕) andanoperation ∗ : R × M → M (called scalar multiplication, usually just written by juxtaposition, i.e. as rx instead of r ∗ x for r ∈ R and x ∈ M) such that for all r, s ∈ R, x, y ∈ M,wehave (a) r(x ⊕ y) = rx ⊕ ry, (b) (r + s)x = rx ⊕ sx, (c) (rs)x = r(sx), (d) 1 x = x where 1 is the multiplicative identity of the ring R. R R From [6,7] we recall the following definition in the fuzzy linear space. Definition 2.4: A fuzzy subset X of a R-module M is called a fuzzy submodule of M if for all x, y ∈ M and r ∈ R,wehave. (a) X(x ⊕ y) ≥ min{X(x), X(y)}. (b) X(rx) ≥ X(x). Remark 2.5: If X is a fuzzy submodule of M, then from b) in Definition 2.4 follows (∀ x ∈ M)(X(0) ≥ X(x)). Definition 2.6: Let A be a fuzzy subset of a nonempty set M.For t ∈ [0, 1], the sets A ={x ∈ M : A(x) ≥ t} and A ={x ∈ M : A(x) ≤ t} are called the upper t-level cut and lower t-level cut of A, respectively. Proposition 2.7: Let M be a R-module. A nonempty subset N of M is a submodule of M if and only if the characteristic function of N is a fuzzy submodule. 420 S. ATAMEWOUE TSAFACK ET AL. Proposition 2.8: A is a fuzzy submodule of a R-module M if and only if for all α, β ∈ R, x, y ∈ M, we have A(αx ⊕ βy) ≥ min{A(x), A(y)}. Proof: The proof is similar to the one for fields in [4], just change a field by ring. Definition 2.9: A fuzzy subset I of a ring R is called a fuzzy ideal of R if for each x, y ∈ R: (a) I(x − y) ≥ min{I(x), I(y)}. (b) I(x · y) ≥ max{I(x), I(y)}. Let G be a group and R a ring. We denote by RG the set of all formal linear combinations of the form α = a g (where a ∈ R and a = 0 almost everywhere, that is only a finite g g g g∈G number of coefficients are different from zero in each of these sums). Let α = a g and β = b g in RG. We define their sum in RG componentwise g g g∈G g∈G by: α + β = (a + b )g and their product by: αβ = a b gh. g g g g∈G g,h∈G With the operations above, RG is a unitary ring, with 1 = u g where the coefficient g∈G corresponding to the unit element of the group is equal to 1 and u = 0 for every other element g ∈ G. We can also define a product of elements in RG by elements λ ∈ R as λ( a g) = g∈G (λa )g. g∈G Definition 2.10: The set RG with the operations defined above is called the group ring of G over R,if R is commutative, then RG is called the group algebra of G over R. Definition 2.11: Let A be a fuzzy subset of the group ring (RG) which is the group algebra of x over the ring Z k, note that x is an invertible element of Z k. If for all α, β ∈ RG, p p (a) A(α.β) ≥ max{A(α), A(β)}; (b) A(α − β) ≥ min{A(α), A(β)}, then A is call a fuzzy ideal of RG. Proposition 2.12: A is a fuzzy ideal of RG if and only if for all t ∈ [0, 1], if A =∅, then A is an t t ideal of RG. Proof: The proof follows from the transfer principle in [8]. Definition 2.13: Let n be an integer. A linear code of length n over Z k is a submodule of Z . In contrast to vector spaces, modules do not admit a basis in general. However modules possess a generating family and therefore a generating matrix, but the decomposition of the elements with respect to this family is not necessarily unique. Definition 2.14: A generating matrix of some linear code over Z is a matrix in M(Z ), k k p p where the lines are the minimal generating family of code. FUZZY INFORMATION AND ENGINEERING 421 Definition 2.15: Let C k and C be two linear codes over Z k with generating matrices G p p and G respectively. The codes C k and C are equivalent if there exists a permutation matrix P,suchthat G = GP (the product of the two matrices G and P). Definition 2.16: Let C be a linear code of length n overZ ,the dual of the code C which k k k p p p ⊥ n ⊥ we denote by C is the submodule of Z defined by: C ={a| for all b ∈ C , a, b = 0}, k k k p p p p where ,  is the inner product. Remark 2.17: C is also a linear code over Z k. Definition 2.18: A linear code C k of length n over Z k is cyclic if it is invariant by the shift p p map s, define by s((a , ... , a )) = (a , a , ... , a ). i.e. if (a , ... , a ) ∈ C , then 0 n−1 n−1 0 n−2 0 n−1 s((a , ... , a )) ∈ C . 0 n−1 3. Fuzzy Linear Code Over Z In this section are defined and studied fuzzy linear linear code over the Galois ringZ . There are also characterised by using the transfer principle [8]. Definition 3.1: Let M = Z be a Z -module. A fuzzy submodule A of M is called a fuzzy k p linear code of length n over Z k. Example 3.2: Let Z be a ring, then Z is a Z -module. 4 4 4 Let A : Z → [0, 1] be the map such that A(0) = A(1) = A(2) = A(3) = t (t ∈ [0, 1]), then A is a fuzzy subset ofZ .Itisobviousthat A is a fuzzyZ -module. Therefore A is a fuzzy linear 4 4 code over Z . Proposition 3.3: Let A be a fuzzy subset of Z . A is a fuzzy linear code of length n over Z if and only if for any t ∈ [0, 1], if A =∅, then A t t is a linear code of length n over Z . Proof: Use the transfer principle in [8]. Corollary 3.4: Let C be a subset of Z . C is a linear code of length n overZ if and only if the characteristic function χ of C is a fuzzy linear code over Z . Proposition 3.5: Let A be a fuzzy subset of Z . A is a fuzzy linear code of length n over Z k if and only if the characteristic function of any upper t-level cut A =∅ for t ∈ [0, 1] is a fuzzy linear code of length n over Z k. Proof: Uses Proposition 3.3 and Corollary 3.4. Remark 3.6: We remark the following: (i) In the Example 3.2, A = Z (for all t ∈ [0, 1]) which is a linear code over Z . t 4 4 422 S. ATAMEWOUE TSAFACK ET AL. (ii) If M is a module over the ring Z and A a fuzzy linear code A on M such that ∀ x ∈ M, A(x) = t (where t ∈ [0, 1]), then A is called the trivial fuzzy linear code over Z . Example 3.7: Consider a fuzzy subset A of Z as follows: 1if x = 0; ⎪ 1 if x = 1; A : Z → [0, 1], x → ⎪ if x = 2; if x = 3. Then A is a fuzzy submodule of the Z -module Z , hence A is a fuzzy linear code over Z . 4 4 4 Remark 3.8: Let A be a fuzzy linear code of length n over Z k.Since Z is a finite set, the image Im(A) ={A(x) | x ∈ Z } is finite as well. Assume that all elements in Im(A) satisfy: t > t > ··· > t (where t ∈ [0, 1]) i.e. Im(A) has m elements. Since A is a linear code over 1 2 m i t Z k,let G be its generator matrix. Thus A can be determined by m matrixes G , G , ... , G t t t t p i 1 2 m (see Theorem 4.7 ). Definition 3.9: Let S be a non-empty set. Let x : S → [0, 1] be a fuzzy subset of S, (where x ∈ S, t ∈ [0, 1]) defined by: t if x = y; x (y) = , for all y ∈ S. 0if x = y. x is called a fuzzy singleton or fuzzy point of S. Definition 3.10: Let A and A be two fuzzy linear codes over Z of thesamelength n. 1 2 The residual quotient of A and A denoted by (A : A ) is the fuzzy subset of Z defined 1 2 1 2 by: (A : A )(r) = sup{t ∈ [0, 1] | r A ⊆ A } for all r ∈ Z . 1 2 t 2 1 That is (A : A ) ={r | r A ⊆ A , r is fuzzy singleton of Z }. 1 2 t t 2 1 t Theorem 3.11 ([2]): Let A and A be two fuzzy linear codes of length n over Z , then the 1 2 residual quotient (A : A ) of A and A is a fuzzy ideal of Z . 1 2 1 2 Definition 3.12: Let A and B be two fuzzy submodules of a module M = Z over the ring Z . We say that A is orthogonal to B if Im(B) ={1 − c | c ∈ Im(A)} and for all t ∈ [0, 1], B = 1−t (A ) ={y ∈ M | x, y = 0, for all x ∈ A }, where ,  is the inner product on M t t Let A and B be two fuzzy submodules of a module M. We denote A orthogonal to B by A ⊥ B. FUZZY INFORMATION AND ENGINEERING 423 Example 3.13: Consider the fuzzy submodules A and B of Z defined as follows: ⎧ ⎧ 1 3 ⎪ ⎪ ⎪ ⎪ if x = 0; if x = 0; ⎪ ⎪ ⎪ ⎪ 2 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ ⎨ if x = 1; ⎨ if x = 1; 4 2 A : Z → [0, 1], x → and B : Z → [0, 1], x → 4 4 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ if x = 2; ⎪ if x = 2; ⎪ ⎪ ⎪ ⎪ 3 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎩ ⎩ if x = 3. if x = 3. 4 2 We have: A ={0} and B = Z , 1/2 1/2 4 A = Z and B ={0}, 1/4 4 3/4 A ={0, 2} and B ={0, 2}. 1/3 2/3 Therefore A ⊥ B. Remark 3.14: Let A be a fuzzy submodule of a module M such that for all x ∈ M, A(x) = γ (with γ ∈ [0, 1]), then it does not exist a fuzzy subset B of M such that A ⊥ B. The Remark 3.14 shows that the orthogonal of some fuzzy submodule does not always exist, so it is important to see under which conditions the orthogonal of a fuzzy submod- ule exists. The following theorem shows the existence of the orthogonal of some fuzzy submodule. Theorem 3.15: Let A be a fuzzy submodule of a finite module M = Z . Then there exists a fuzzy submodule B of M such that A ⊥ B if and only if |Im(A)| > 1 and for any γ ∈ Im(A) there exist  ∈ Im(A) such that A = (A ) . Proof: Let A be a fuzzy submodule of M. Assume that |Im(A)|= m > 1 and for any γ ∈ Im(A) there exists  ∈ Im(A) such that A = (A ) . Assume that Im(A) ={t > t > ··· > t }. Define the sets M ={x ∈ M | A(x) = t },for 1 2 m i i i = 1, ... , m. These sets form a partition of M. We define a fuzzy set B as follows: B : M → [0, 1], x → 1 − t ,if x ∈ M . m−i+1 i Since Im(A) ={t > t > ··· > t },wehave A ⊆ A ⊆ ··· ⊆ A .Asforany γ ∈ Im(A) 1 2 m t t t 1 2 m there exists  ∈ Im(A) such that A = (A ) , with the properties that we know about the orthogonal over the finite module, we conclude that A = (A ) .Thus B = t t 1−t i m−i+1 m−i+1 {x ∈ M | B(x) ≥ 1 − t }= M ∪ M ∪ ··· ∪ M = A = (A ) .Thus B is the fuzzy m−i+1 i i−1 1 t t i m−i+1 submodule we need. Conversely if there exists a fuzzy submodule B of M such that A ⊥ B, then by Definition 3.12, |Im(A)| > 1. Since ∀ t ∈ [0, 1], B = (A ) , then ∀γ ∈ Im(A), there exist 1−t t ∈ Im(A),suchthat A = (A ) ,because Im(B) ={1 − t | t ∈ Im(A)}. The following result shows the uniqueness of the orthogonal of some fuzzy submodule. Theorem 3.16: Let A, B and C be three fuzzy submodules of a module M, such that A ⊥ Band A ⊥ C, then B = C. 424 S. ATAMEWOUE TSAFACK ET AL. Proof: Assume that A ⊥ B and B ⊥ C.Let t ∈ [0, 1], and y ∈ B . Then x, y = 0, for all x ∈ 1−t A .Thus y ∈ C and B ⊆ C . Therefore C ⊆ B . In the same way, we show that B ⊆ C . t 1−t 1−t 1−t t t t t Therefore B = C. Corollary 3.17: Let A be a fuzzy submodule of a finite module M such that there exists a fuzzy set B on M orthogonal to A, then B is a fuzzy submodule of M. ⊥ ⊥ ⊥ Corollary 3.18: Let A be a fuzzy submodule of M. If A exists, then (A ) = A. Definition 3.19: Let A and B be two fuzzy linear codes overZ . A and B are equivalent fuzzy linear codes over Z if for all t ∈ [0, 1], the linear codes A and B are equivalent. t t Example 3.20: (1) All fuzzy linear code are equivalent to itself. (2) Let C and C be two equivalent linear codes of length n over Z . We define two G G k 1 2 p equivalent fuzzy linear codes as follows: 1if x ∈ C ; A : Z → [0, 1], x → 0 otherwise. and 1if x ∈ C ; B : Z → [0, 1], x → 0 otherwise. n n Thus A = C and B = C , A = Z and B = Z . 1 G 1 G 0 0 1 2 k k p p Remark 3.21: Let A and B be two equivalent fuzzy linear codes over Z , then Im(A) = Im(B). Let’s draw the communication channel as follows: k 2 n 3 3 Assume that F = Z and F = Z , that means that k = 2and n = 3. Let C ⊆ F be a 2 2 linear code over F, in the classical case, when we send a codeword c = (101) ∈ C through a communication channel, the signal receive can be read as c= (0.98, 0.03, 0.49) and mod- ulate to c = (100). Thus to know if c belong to the code C, we use syndrome calculation [9]. Since the modulation have gave a wrong word, we can consider that c have more infor- mation than c , in the sense that we can estimate a level to which a word 0 is modulate to 1, and a word 1 is modulate to 0. Therefore it is possible to use the idea of fuzzy logic to recover the transmit codeword. Let a linear code C ⊆ Z .Toeach c ∈ C, we find t ∈ [0, 1] such that t estimate the degree of which the element of R ,obtainfrom c through the transmission channel belong to the 3 3 code C.ThusinZ the information that we handle are certain, whereas inR there are uncer- tain. When we associate to all elements of Z the degree of which its correspond element obtain through the transmission channel belong to Z , then we obtain a fuzzy code. If the fuzzy code are fuzzy linear code, then we can recover the code C just by using the upper t-level cut. Thus we deal directly with the uncertain information to obtain the code C. FUZZY INFORMATION AND ENGINEERING 425 The following example illustrate this reconstruction of the code by using uncertain information in the case of fuzzy linear code. Example 3.22: LetZ ={000, 001, 010, 100, 110, 101, 011, 111} and C ={000, 001, 110, 111} be a linear code over Z . Assume that after the transmission we obtain respectively {000; 0.01, 01; 1.01, 10; 1.001, 1, 0.999}.Let ⎪ {1} if x = 000; {0.99} if x = 001; {0.9} if x = 010; {0.9} if x = 100; A : Z → [0, 1] such that x → {0.99} if x = 110; ⎪ {0.9} if x = 101; ⎪ {0.9} if x = 011; {0.99} if x = 111. Then by finding a t ∈ [0, 1] such that A ={x ∈ Z | A(x) ≥ t}= C,weobtain t > 0.9. Thus, for t = 0.99, we are sure that the receive codeword is in C. 4. Fuzzy Cyclic Codes Over Z k In this section, we will study the case where the integers n and p are coprime. Definition 4.1: A fuzzy submodule A of the module Z is called a fuzzy cyclic code of length n over Z k if for all (a , a , ... , a ) ∈ Z ,wehave A((a , a , ... , a )) ≥ 0 1 n−1 n−1 0 n−2 A((a , a , ... , a )). 0 1 n−1 Proposition 4.2: Let A be a fuzzy submodule A of the module Z . A is a fuzzy cyclic code on n n Z if and only if for all t ∈ [0, 1], if A =∅, then A is a cyclic code on Z . t t k k p p Proof: The proof uses the transfer principle from [8]. As well as the Proposition 3.5 in the linear case, we have the following result in the cyclic case. Proposition 4.3: Let A be a fuzzy module A on the module Z . A is a fuzzy cyclic code on Z if and only if the characteristic of any upper t-level cut A =∅ for t ∈ [0, 1] is a fuzzy cyclic code on Z . n n Proposition 4.4: A is a fuzzy cyclic code on Z if and only if for all (a , a , ... , a ) ∈ Z , 0 1 n−1 k k p p then A((a , a , ... , a )) = A((a , a , ... , a )) = ··· = A((a , a , ... , a , a )). 0 1 n−1 n−1 0 n−2 1 2 n−1 0 426 S. ATAMEWOUE TSAFACK ET AL. Proof: Assume that A is a fuzzy cyclic code on Z . Then A((a , a , ... , a )) ≤ A((a , a , ... , a )) ≤ ··· ≤ A((a , a , ... , a , a )) 0 1 n−1 n−1 0 n−2 1 2 n−1 0 ≤ A((a , a , ... , a )). 0 1 n−1 Therefore A((a , a , ... , a )) = A((a , a , ... , a )) = ··· = A((a , a , ... , a , a )). 0 1 n−1 n−1 0 n−2 1 2 n−1 0 The converse is straightforward by Definition 4.1. Proposition 4.5: A is a fuzzy cyclic code of length n over Z if and only if for all t ∈ [0, 1], if A =∅, then A is an ideal of the factor ring Z [X]/(X − 1). t t n n Proof: Let φ be a mapping defined in [5] as follows, φ : Z → Z [X]/(X − 1),suchthat k p n−1 c = (c , ... , c ) → φ(c) = c X . φ is an isomorphism of Z -module, which sends a 0 n−1 i k i=0 p cyclic code over Z onto the ideals of the factor ring Z [X]/(X − 1). k k p p Let A be a fuzzy subset of Z . Assume that A is a fuzzy cyclic code over Z k. Let t ∈ [0, 1] such that A =∅, then A is a cyclic code over Z . Therefore, ∀ t ∈ [0, 1], A t t t is an ideal of Z [X]/(X − 1). Conversely, assume that for all t ∈ [0, 1] such that A =∅, A is an ideal of factor ring t t n n Z [X]/(X − 1).Since A is an ideal of factor ring Z [X]/(X − 1), then A is a submodule k k t t p p of Z -module Z . Hence A =∅, is a linear code over Z , then A is a fuzzy linear code. k k p k p Such as φ is define, A is a cyclic code over Z , for all t ∈ [0, 1]. Hence A is a fuzzy cyclic code over Z . Proposition 4.6: A is a cyclic fuzzy code of length n if and only if A is a fuzzy ideal of the group algebra RG, which is the group algebra of x over the finite ring Z . Proof: Let A be a fuzzy cyclic code. For any α, β ∈ RG, we have A(α − β) ≥ min{A(α), A(β)} since A is a fuzzy Z k-module on Z . 2 n−1 If α ∈ RG, then A(xα) ≥ A(α), A(x α) ≥ A(α), ··· , A(x α) ≥ A(α).Sofor β ∈ RG, n−1 i n−1 with β = l x ,(l ∈ Z k) we conclude A(α.β) = A(l α + l xα + ··· + l x α) ≥ i i 0 1 n−1 i=0 n−1 min{A(l α), ... , A(l x α)}≥ A(α). Similarly we can also show that A(α.β) ≥ A(β). 0 n−1 Hence A(α.β) ≥ max{A(α), A(β)}. Conversely, assume that A is a fuzzy ideal of the group algebra RG. (Note that RG is a module that has G = x as base). Since A is a fuzzy ideal of RG, also A is a fuzzy submodule of RG. n n−1 For any (a , ... , a ) ∈ Z we associate α = (a + a x + ··· + a x ). Then 0 n−1 0 1 n−1 n−1 n−1 A(a + a x + ··· + a x ) = A(x(a + a x + ··· + a x )) ≥ max{A(x), A(a + n−1 0 n−2 0 1 n−1 0 n−1 n−1 a x + ··· + a x )}≥ A(a + a x + ··· + a x ). Therefore A((a , a , ... , a )) 1 n−1 0 1 n−1 n−1 0 n−2 ≥ A((a , ... , a )). Hence A is a fuzzy cyclic code. 0 n−1 Since Z k is a finite ring, also Im(A) ={A(x) ∈ [0, 1]|x ∈ Z } is finite. Let Im(A) ={t > (k) t > ··· > t }, then A ⊆ A ⊆ ··· ⊆ A ⊆ A = Z .Let g (X) ∈ Z [X] be the gen- 2 m t t t t 1 2 m−1 m k p (k) erator polynomial of A . Note that g (X) is the Hensel lift of order k of some polynomial i i FUZZY INFORMATION AND ENGINEERING 427 (k) n n g (X) ∈ Z [X] which divides X − 1. The cyclic code g (X)⊂ Z k[X]/(X − 1) is called i p the lift code of the cyclic code g (X)⊂ Z [X]/(X − 1). For more information about Hensel i p lifting see [5]. (k) (k) Since A ⊆ A ⊆ ··· ⊆ A ⊆ A = Z , it follows that g (X)|g (X), i = 1, ... , t t t t 1 2 m−1 m k i+1 i m − 1. (k) (k) We can define the polynomial h (X) = (X − 1)/g (X) whichiscalledthe check i i (k) polynomial of the cyclic code A =g (X), i = 1, ... , m. i i (k) (k) (k) Theorem 4.7: Let G ={g (X), g (X), ... , g (X)} be a set of polynomials in Z [X], 1 2 p (k) (k) such that g (X) divide X − 1, i = 1, ... , m. If g (X) | g (X) for i = 1, 2, ... , m − 1 and i+1 i (k) (k) g (X)= Z , then the setG determines a fuzzy cyclic code A and {g (X)| i = 1, ... , m} is the family of upper level cut cyclic subcodes of A. (k) (k) (k) (k) Proof: Since g (X) | g (X),wehave g (X)⊆g (X) for i = 1, 2, ... , m − 1. Choose i+1 i i i+1 t ∈ [0, 1] such that t > t > ··· > t . i 1 2 m (k) Let A =g (X) for i = 1, 2, ... , m − 1. We define A as follows. i i (k) t ,if c ∈g (X); A(c) = (k) (k) t,if c ∈g (X)\g (X), i = 2, ... , m. i i−1 n−1 n n i where φ : Z → Z k[X]/(X − 1), c = (c , ... , c ) → φ(c) = c X is an isomor- 0 n−1 i p i=0 phism of Z -module. (k) Since for all t ∈ [0, 1], A =g (X) is a cyclic code as it is an ideal of the principal ring i t i i (k) Z [X]/(X − 1), i = 1, ... , m,also A is a fuzzy cyclic code and {g (X)| i = 1, ... , m} is the family of upper level cut cyclic subcodes of A. (k) Corollary 4.8: With the same notations and hypothesis as in Theorem 4.7, if g (X) = Z , m k (k) then the setG determines a fuzzy cyclic code A and {g (X)| i = 1, ... , m}∪ Z k[X]/(X − 1) i p is the family of upper level cut cyclic subcodes of A. Proof: Take (k) t ,if c ∈g (X); ⎪ 1 (k) (k) t,if c ∈g (X)\g (X), i = 2, ... , m; i i−1 A(c) = Z [X] (k) 0, if c ∈ \g (X). (X − 1) Proposition 4.9: Let A be the fuzzy cyclic code of length n overZ that can be determined by (k) (k) the set of polynomialG ={g (X) | i = 1, ... , m} as in Theorem 4.7.Ifforallg (X) ∈ G there i i (k) (k) (k) (k) exists g (X) ∈ G such that g (X).g (X) = X − 1, then the set of polynomials {h (X) = j i j i (k) (X − 1)/g (X) | i = 1, ... , m} determines the orthogonal of A. i 428 S. ATAMEWOUE TSAFACK ET AL. Proof: Under the conditions of Theorem 4.7 we define A as follows. (l) 1 − t ,if c ∈h (X); 1 m A (c) = (l) (l) 1 − t,if c ∈h (X)\h (X), i = 2, ... , m. i−1 i ⊥ ⊥ Since the upper level set (A ) is a linear code, A is the orthogonal of A. Theorem 4.10: Let A and A be two fuzzy cyclic codes on Z , then: 1 2 (i) A ∩ A is a fuzzy cyclic code, 1 2 (ii) A + A is a fuzzy cyclic code, 1 2 (iii) A A is a fuzzy cyclic code. 1 2 Proof: Let A and A be two fuzzy cyclic codes of the Z -module Z . 1 2 k (i) Let (a , a , ... , a ) ∈ Z . 0 1 n−1 A ∩ A ((a , a , ... , a )) 1 2 n−1 0 n−2 = min{A ((a , a , ... , a ), A ((a , a , ... , a ))} 1 n−1 0 n−2 2 n−1 0 n−2 ≥ min{A (a , a , ... , a ), A ((a , a , ... , a ))} 1 0 1 n−1 2 0 1 n−1 = A ((a , a , ... , a )) ∩ A ((a , a , ... , a )). 1 0 1 n−1 2 0 1 n−1 Since the intersection of two fuzzy modules is a fuzzy module, we obtain that A ∩ A 1 2 is a fuzzy cyclic code over Z k. (ii) For all (a , ... , a ) ∈ Z ,wehave: 0 n−1 (A + A )((a , a , ... , a )) 1 2 n−1 0 n−2 = max{A ((b , b , ... , b )) ∧ A ((c , c , ... , c )) | b 1 n−1 0 n−2 2 n−1 0 n−2 i + c = a , i = 0, ... , n − 1} i i ≥ max{A ((b , b , ... , b )) ∧ A ((c , c , ... , c ))|b + c = a , i = 0, ... , n − 1} 1 0 1 n−1 2 0 1 n−2 i i i = (A + A )((a , a , ... , a )). 1 2 0 1 n−1 Since A + A is a fuzzy module, we conclude as above that A + A is a fuzzy cyclic 1 2 1 2 code. (iii) It is similar to A + A . 1 2 5. Fuzzy Z -linear Codes After having studied the notion of fuzzy linear code over the ringZ k in the previous section, we are now going to construct fuzzy Z k-linear codes explicitly. 5.1. Fuzzy Gray Map Initially, the code of Gray is an order on the binary sequences of a fixed length n, permitting to enumerate all these sequences while modifying only one bit in order to pass from one FUZZY INFORMATION AND ENGINEERING 429 sequence to the next one. The case that is going to interest us directly is the one of the sequence of length two, for which one has the following Gray code: 0 → 00 1 → 01 2 → 11 3 → 10. Let ψ : Z 2 → Z the Gray map. We are going to define the fuzzy Gray map between two 2 2 fuzzy spaces by the extension principle [10]. 2 2 Definition 5.1: Let ψ : Z → Z be the Gray map, and let F (Z ), F (Z ) be the set of 2 2 2 2 2 2 all fuzzy subsets of Z and Z respectively. The fuzzy Gray map is the map ψ : F (Z ) → 2 2 2 2 F (Z ), such that for any A ∈ F (Z ), ψ(A)(y) = sup{A(x) | y = ψ(x)}. 2 2 Example 5.2: Let 1if x = 0; ⎪ 1 if x = 1; A : Z → [0, 1], x → if x = 2; if x = 3. with the Gray map ψ(0) = 00, ψ(1) = 01, ψ(2) = 11, ψ(3) = 10. 1 1 By the fuzzy Gray map we have ψ(A)(00) = 1, ψ(A)(01) = , ψ(A)(11) = , ψ(A)(10) = 3 3 , hence ψ(A) is a fuzzy linear code. Theorem 5.3: The fuzzy Gray map ψ is a bijection. Proof: This follows from the fact that ψ is a one to one function. As in crisp case, we have the following proposition. Proposition 5.4: If A is a fuzzy linear code over Z 2 and ψ the Gray map, then ψ(A) is not always a fuzzy linear code over the field Z The Gray map allows to construct nonlinear codes as binary images of the linear codes, that is the case of Kerdock, Preparata, and Goethals codes. For a good understanding, we suggest the reader to examine [11,12]. In fact if C is a linear code of length n over Z , then C = ψ(C) is a nonlinear code of length 2n over Z in general [11]. In that way we construct a fuzzy Kerdock code in the following example. 430 S. ATAMEWOUE TSAFACK ET AL. Example 5.5: Let C be a linear code of length 8 over Z with the generator matrix ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ G = , ⎝ ⎠ then its image under the Gray map ψ gives a Kerdock code C (see construction in [9]). Let 1, if x ∈ C; A : Z → [0, 1], x → 0, otherwise. Then A is a fuzzy linear code over Z . Since ψ is a bijection, we construct 1, y ∈ E; ψ(A) : Z → [0, 1], y → 0, otherwise, where E ={y ∈ Z | y = ψ(x) and x ∈ C} Since E is not a linear code over Z ,weconcludethat ψ(A) is a fuzzy Z -linear code but 2 2 not a fuzzy linear code Z . Consequently, ψ(A) is a fuzzy Kerdock code of length 16. Remark 5.6: A fuzzy Z -linear code is not in general a fuzzy linear code over Z 4 2 If we define the fuzzy binary relation R on Z 2 × Z by 1, if y = ψ(x); R (x, y) = 0, otherwise. It is easy to see [13]that ψ(A)(y) = sup{A(x) | y = ψ(x)} can be represented by ψ(A)(y) = sup{min{A(x), R (x, y)}| x ∈ Z }. k−1 As in [5], let : Z → Z be the generalised Gray map. k−1 Definition 5.7: We call the map : F (Z k ) → F (Z ), such that for any A ∈ F (Z k ), p p sup{A(x) | y = (x)},ifasuch x exists; (A)(y) = 0, otherwise. The fuzzy generalised gray map. k−1 Since : Z → Z cannot give more than one image for one element, then Definition 5.7 can be simply write A(x),if y = (x); (A)(y) = 0, otherwise. k−1 k−1 p p Remark 5.8: Let B ∈ F (Z ) such that B(y) = t = 0 for any y ∈ Z . There does not p p exist a fuzzy subset A ∈ F (Z k ) such that (A) = B.Thus is not a bijection map. p FUZZY INFORMATION AND ENGINEERING 431 5.2. Fuzzy Z k-linear Codes k−1 n.p In the following, we will denote by the map from F (Z ) onto F (Z ) which spreads k p the fuzzy generalised Gray map. Definition 5.9: A fuzzy code A over Z is a fuzzy Z -linear code if it is an image under the p k fuzzy generalised Gray map of a fuzzy linear code over the ring Z . Definition 5.10: A fuzzy code A is a fuzzy Z k-cyclic code if it is a fuzzy Z k-linear code and p p if it is the image under the generalised Gray map of a cyclic code over the ring Z . Remark 5.11: A fuzzy Z -linear code is a fuzzy code over the field Z . Example 5.12: (1) Let 1, if e = f = 0; B : Z → [0, 1], w = (a, b, c, d, e, f ) → 0, otherwise. B is a fuzzy linear code of length 6 over Z .Let 1, if z = 0; A : Z → [0, 1], v = (x, y, z) → 0, otherwise. A is a fuzzy linear code of length 3 over Z . Moreover, if B = ψ(A), then B is a fuzzy Z -linear code. (2) Let B : Z → [0, 1], ,if d = e = f = g = h = i = 0and abc ∈ {000, 012, 021, 111, 120, 102, 222, 201, 210}; ⎨ 1 ,if g = h = i = 0, abc ∈{000, 012, 021, 111, v = (a, b, c, d, e, f, g, h, i) → 120, 102, 222, 201, 210} and def ∈{012, 021, 111, 120, 102, 222, 201, 210}; 0, otherwise. B is a fuzzy linear code over Z and B is a fuzzy Z 2-linear code. It is easy to show the next proposition, whose Example 5.5 is a perfect illustration of it. Proposition 5.13: Let B be a fuzzy Z -linear code, then B is not always a fuzzy linear code over the field Z . Since the fuzzy generalised Gray map image of fuzzy linear code is a fuzzy codes over the field Z , we can also construct fuzzy Z -linear codes using the following diagram: p 432 S. ATAMEWOUE TSAFACK ET AL. Example 5.14: (1) Let A : Z → [0, 1] be a linear code such that A has three upper t- level cuts A ⊆ A ⊆ A .Let A = (A ), A = (A ) and A = (A ),wehave A = t t t t t t 3 2 1 t 3 t 2 t 1 t 3 2 1 3 (A ) ⊆ A = (A ) ⊆ A = (A ). We construct A = (A) as follow. t t t 3 t 2 t 1 2 1 ⎪ t ,if y ∈ A ; t ,if y ∈ A ; k−1 n.p 2 A : Z → [0, 1], y → t ,if y ∈ A ; ⎪ t 0, otherwise. (2) Let if x = 0; A : Z → [0, 1], x → if x = 2; if x = 1, 3, be a fuzzy linear code over Z . Then A ={0}, A ={0, 2} and A = Z . 4 1/2 1/3 1/4 4 We construct A ={00}, A ={00, 11} and A = Z , the Gray map image of A , 1/2 1/2 1/3 1/4 2 A and A respectively, we define 1/3 1/4 ⎪ if x ∈ A , y = ψ(x); 1/2 A : Z → [0, 1], y → if x ∈ A \ A , y = ψ(x); 1/3 1/2 if x ∈ A \ A , y = ψ(x). 1/4 1/3 Remark 5.15: A and ψ(A) are the same codes. Proposition 5.16: If for all t ∈ [0, 1], A = (A )(when A = 0) is a linear code over Z , then t t p these two constructions of fuzzy Z -linear codes above give the equivalent fuzzy codes. Proof: This follows directly from the definition of the fuzzy generalised Gray map and the fact that the image under the generalised Gray map of a linear code is not a linear code in general.  FUZZY INFORMATION AND ENGINEERING 433 6. Conclusion In this paper where we study fuzzy coding, we define fuzzy linear codes over the finite ring Z , fuzzy generalised Gray map and fuzzy Z -linear codes. We also investigate some of k k p p their properties and remark that many of them are similar to the classical form. The codes of Kerdock are permit us to show that fuzzy Z -linear codes is not a fuzzy linear code. This work allows us to reinforce the hypothesis of Von Kaenel [14], that the theory of fuzzy sets is a natural setting for this study. Acknowledgments We are grateful to the referee for their valuable suggestions, which have improved this paper. Disclosure statement No potential conflict of interest was reported by the authors. Notes on contributors S. Atamewoue Tsafack is a member of Research Team in Algebra and Logic (ERAL) in the Department of Mathematics in the Faculty of Science at the University of Yaounde I, Cameroon. I works on General Algebra, Fuzzy Logic and Coding Theory. S. Ndjeya is a Senior lecturer in the Department of Mathematics in the Faculty of Science at the University of Yaounde I, Cameroon. He works on Pure and Applied Algebra, and Coding Theory. L. Strüngmann is full Professor in the Faculty of Computer Science Institute of Applied Mathematics at the Mannheim University of Applied Sciences, in Germany. He works on Coding Theory, General Algebra and Bioscience. C. Lele is Full Professor in the Department of Mathematics and Computer Sciences in the Faculty of Science at the University of Dschang, Cameroon. He works on General algebra and Fuzzy logic. References [1] Zadeh LA. Fuzzy sets. Inf Control. 1965;8:338–353. [2] Negoita CV, Ralescu DA. Applications of fuzzy sets and system analysis. Basel and Stuttgart: Birkhäuser verlag; 1975. 191PP. [3] Maschinchi M, Zahedi MM. On L-fuzzy primary submodules. Fuzzy Sets Syst. 1992;49:231–236. [4] Shum KP, De Gang C. Some note on the theory of fuzzy code. BUSEFAL (Bulletin Pour Les Sous Ensembles Flous et Leurs Applications) University of Savoie, France. 2000;81:132–136. [5] Galand F. Construction de codes Z k -linéaires de bonne distance minimale et schémas de dissimulation fondés sur les codes de recouvrement [PhD thesis]. Université de Caen; 2004. [6] Biswas R. Fuzzy fields and fuzzy linear spaces redefined. Fuzzy Sets Syst. 1989;33:257–259. [7] Nanda S. Fuzzy fields and fuzzy linear space. Fuzzy Sets Syst. 1986;19:89–94. [8] Kondo M, Dubek WA. On transfer principle in fuzzy theory. Mathware Soft Comput. 2005;12:41–55. [9] Carlet C. Z -linear codes. IEEE Trans Inf Theory. 1998;44:1543–1547. [10] Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning I, II, III. Inf Sci. 1975;8–9:199–249, 301–357, 43–80. [11] Hammons R, Kumar PV, Calderbank AR, et al. Kerdock, Preparata, Goethals and other codes are linear over Z . IEEE Trans Inf Theory. 1994;40:301–319. [12] Kerdock AM. A class of low-rate nonlinear binary codes. Inf Control. 1972;20:182–187. 434 S. ATAMEWOUE TSAFACK ET AL. [13] Perfilieva I. Fuzzy Function: Theoretical and Practical Point of View. Proceedings of the 7th con- ference of the European Society for Fuzzy Logic and Technology, Aix-les-Bains, France: Atlantis Press; 2011. pp. 480–486. Available from: https://doi.org/10.2991/eusflat.2011.142. [14] Von Kaenel PA. Fuzzy codes and distance properties. Fuzzy Sets Syst. 1982;8:199–204.

Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Oct 2, 2018

Keywords: Code over Galois ring; fuzzy linear code; fuzzy cyclic code; fuzzy generalised gray map

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