A note on the symbol - Pair distance of repeated-root negacyclic codes of length 14

Southeast Asian J. of Sciences: Vol. 6, No. 1 (2018) pp.17-27 A NOTE ON THE SYMBOL-PAIR DISTANCE OF REPEATED-ROOT NEGACYCLIC CODES OF LENGTH 14 Nguyen Trong Bac Department of Basic Sciences University of Economics and Business Administration Thai Nguyen University, Thai Nguyen 250000, Vietnam e-mail: bacnt2008@gmail.com Abstract In this paper, the symbol-pair distances of all repeated-root negacyclic codes of length 14 are obtained. As an application, all MDS symbol-pair negacyclic codes of length 14 over finite field F7m are established. 1. Introduction In [1], Shannon showed that good codes exist, gave birth to information theory and coding theory. Although its origins is to solve the problem about reliable communication in an engineering problem, the subject has developed by using more and more mathematical techniques. Cyclic codes are the most studied of all codes, since their rich algebraic structures and practical implementations. Moreover, cyclic codes can build blocks for many other codes, such as BCH, Kerdock, Preparata and Justesen codes. Negacyclic codes as a direct gener- alization of cyclic codes, they also have rich algebraic structure and can be efficiently encoded using shift registers, have attracted remarkable attention for the last half of the century. Let Fpm be the finite field of order pm, where p is an odd prime. Nega- cyclic codes of length n over Fpm are defined by the ideals〈g(x)〉 of quotient Key words: Negacyclic codes, repeated-root codes, dual codes, Hamming distance, symbol- pair distance, MDS codes. 17 18 A note on the Symbol-Pair Distance of... ring Fpm〈xn+1〉 , where the generator polynomial 〈g(x)〉 is the monic polynomial of least degree in the code, and is a divisor of xn +1. In fact, all previous studies, most researcher focus their attention on the situation that gcd(n, p) = 1. This is equivalent to say that generator polynomial g(x) has no repeated irreducible factors, these codes are called simple-root negacyclic codes. Instead, when gcd(n, p) = n, i.e., generator polynomial g(x) has repeated roots in an exten- sion filed. We called these codes are repeated-root codes, which were first inves- tigated in the 1990s by Castagnoli in [7] and Van Lint in [37]. In these papers, the authors have shown that repeated-root cyclic codes cannot be asymptoti- cally better than simple-root cyclic codes. But, there still exist a few optimal such codes (see, for example, [12 14]), which encourages many researchers to study the class of codes. The reader can refer to [16, 17, 18, 5, 24, 11, 12]. Let Σ be an alphabet of size q, whose elements are called symbols. Suppose that x = (x0, x1, · · · , xn−1) is a vector in Σn, in [8], Cassuto and Blaum defined the symbol-pair vector of x as πsp(x) = [(x0, x1), (x1, x2), · · · , (xn−2, xn−1), (xn−1, x0)] ∈ (Σ2)n. (1) Two pairs (c, d) and (e, f) are distinct if c = e or d = f , or both. For any two vectors x and y, the symbol-pair distance between x and y is defined as dsp(x,y) = dH(πsp(x), πsp(y)), where dH denotes the usual Hamming dis- tance. Accordingly, if the pair (c, d) = (0, 0), we say wtH(c, d) = 1, otherwise, wtH(c, d) = 0. Then the symbol-pair weight of a vector x is defined as wtsp(x) = wtH(πsp(x)) = ∣∣∣{i | (xi, xi+1) = (0, 0), 0 ≤ i ≤ n− 1, xn = x0}∣∣∣. A q-ary code C of length n over Σ can be regard as a nonempty subset of Σn. Then the minimum symbol-pair distance of a code C is defined to be dsp(C) = min(dsp(c1, c2) | c1, c2 ∈ C, c1 = c2). Clearly, if C is a linear pair code, the symbol-pair weight and minimum symbol-pair distance are the same, that is to say, dsp(C) = min(wtsp(c)|c = 0, c ∈ C). In [8, 9], Cassuto and Blaum has shown that the symbol-pair codes are designed to protect against pair errors in symbol-pair read channels, and a symbol-pair code C can correct t pair-errors if and only if dsp > 2t + 1, where dsp is the minimum pair-distance of C. So the minimum pair-distance is one of the important parameters of a symbol-pair code. For any code C of length n with 0 < dH < n, a simple but important connection between dH and dsp is given in [8]: dH + 1 ≤ dsp ≤ 2dH. Later on, in [10], Cassuto and Litsyn have shown that for any cyclic code C of length n with Hamming distance dH, if the generator polynomial g(x) of C has at least dH roots in the splitting field of xn − 1, then the symbol-pair distance of C is at least dH + 2. In addition, in [10], they proved that if the length n of cyclic code is prime, and N. T. Bac 19 the generator polynomial g(x) of C has at least m roots in the splitting field of xn − 1, and dH ≤ min{2m − n + 2, m− 1}, then the symbol-pair distance of C is at least dH + 3. Recently, in [38], Yaakobi et al. considered the lower bound of binary cyclic code and showed the result: for any linear cyclic code of dimension greater than one with a minimum Hamming distance dH, the symbol-pair distance is at least dH +dH2 . It is well known for any fixed code length n and dimension k, maximum distance separable(MDS) code has the largest minimum distance, i.e., they have the best possible error-correction capability. Thus, how to construct MDS code always a hot topic in coding theory. As a generalization of MDS codes, MDS symbol-pair codes also have the best possible error-correction capability. More recently, in[15], Chee et al. established singleton Bound for symbol-pair codes as follows: Let 2 ≤ dsp ≤ n, then for any symbol-pair code C of length n with size M and minimum pair-distance dsp over Fq , M ≤ qn−dsp +2. If the equality hold then the symbol-pair code C is called an optimal code with respect to Singleton bound, or MDS symbol-pair code. After establishing the Singleton Bound, a lot of work focus on how to construct MDS symbol-pair codes(see, for example, [15, 14]). But a few work has been done on how to determine the symbol-pair distance of some classes of linear code as it is generality difficult to determine. In [4], Dinh et al. computed the symbol-pair distance of all constacyclic codes of length 5 over F7m . As an application, they obtained a lot of MDS symbol-pair codes. Motivated by [4], in this paper, we get the symbol-pair distance of all negacyclic codes of length 14 over F7m and obtained numerous symbol-pair codes. Moreover, we find that our result are also suitable for all constacyclic codes of length 14 over F7m . The remainder of the paper is organized as follows. Section 2 recalls some preliminary results. In Section 3, we study the symbol-pair distance of nega- cyclic codes of length 14. In Section 4, we give the all MDS symbol-pair codes of length 14. 2. Preliminaries In this Section, we state some basic fact about finite ring and constacyclic codes. A principal ring is a ring in which each ideal generated by a single element. A chain ring is a principal ring such that the ideals are linearly orders under set theoretic containments. Let R be a finite ring. An element r ∈ R is said to be nilpotent with nilpotency index l if rl = 0 and l is the least positive integer with respect to this property. It follows that if R is a finite commutative chain ring, then there is an element γ such that γ generator of the unique maximal 20 A note on the Symbol-Pair Distance of... ideal of R. Hence, the ideals of R are 〈γi〉 and they form a chain: R = 〈γ0〉  〈γ1〉  · · ·  〈γl−1〉  〈γl〉 = 〈0〉. Let p be an odd prime, m be a positive integer, and Fpm be a finite field. A code C of length n over Fpm is a nonempty subset of Fnpm . An [n, k]-linear code C over the finite field F7m is a k-dimensional linear subspace of Fnpm . Moreover, For a nonzero element λ of F7m , if (c0, c1, · · · , cn−1) ∈ C implies (λcn−1, c0, · · · , cn−2) ∈ C, then C is called a λ-constacyclic code. It is well known that any constacyclic code C of length n over F7m corresponds to an ideal of Fpm [x]/(xn + λ) and it can be expressed as C = (g(x)), where g(x) is monic and has least degree in the code. In the case λ = −1, those λ-constacyclic codes are called cyclic codes, and when λ = 1, such λ-constacyclic codes are called negacyclic codes. From that, negacyclic codes of length 14 over F7m correspond to the ideals of the finite ring R1 = Fpm [x]〈x10 + 1〉 . Clearly, R1 is a principal ideal ring, whose ideals are generated by factors of x10 +1. In [3], the authors have shown that the polynomial x2 +1 ∈ Fpm [x] is irreducible if and only if pm ≡ 4k+3 for some integer positive m (see Lemma 7.8). Hence, if pm ≡ 3 (mod 4), then the monic divisors of x10+1 = (x2+1)ps are the set {x2 + 1)i : 0 ≤ i ≤ ps}. Therefore, R1 is a chain ring, whose maximal ideal is (x2+1). Similarly, If pm ≡ 1 (mod 4), then the monic divisors of x10 + 1 = (x − γ)ps (x + γ)ps are the set {(x − γ)i(x + γ)j : 0 ≤ i, j ≤ ps}, where γ ∈ Fpm such that γ2 = −1. Therefore, R1 is a principal ideal ring, but not a chain ring, whose maximal ideal is (x− γ) or (x + γ). The following is well known fact about R1. Proposition 1. (Theorem 3.2 of [16]) Let p be an odd prime, and m be a positive integer. (a) If pm ≡ 1 (mod 4), negacyclic codes of length 14 over F7m are 〈(x−γ)i(x+ γ)j〉 ⊆ R1, where 0 ≤ i, j ≤ ps. Each code Ci,j = 〈(x − γ)i(x + γ)j 〉 contains pm(10−i−j) codewords, its dual is C⊥i,j = 〈(x−γ)p s−i(x+γ)p s−j〉. (b) If pm ≡ 3 (mod 4), negacyclic codes of length 14 over F7m are 〈(x2+1)i〉 ⊆ R1, where 0 ≤ i ≤ ps. Each code Ci = 〈(x2 + 1)i〉 contains p2m(ps−i) codewords, its dual is C⊥i = Cps−i = 〈(x2 + 1)p s−i〉. Given two codewords x = (x0, x1, . . . , xn−1),y = (y0, y1, . . . , yn−1) ∈ Fnpm , their inner product is defined as: x · y = x0y0 + x1y1 + · · ·+ xn−1yn−1. N. T. Bac 21 Then x,y are called orthogonal if x · y = 0. For a linear code C over Fpm , its dual code C⊥ is the set of n-tuples over Fpm that are orthogonal to all codewords of C, i.e., C⊥ = {x ∈ Fnpm | x · y = 0, ∀y ∈ C}. In particular, a code C is called self-orthogonal if C ⊆ C⊥ and it is called dual-containing if C⊥ ⊆ C. Moreover, it is called self-dual if C = C⊥. Then, making use of Proposition 1, it is straightforward for us to get the necessary and sufficient conditions for negacyclic codes of length 14 over Fpm to be self-dual, self-orthogonal, dual containing. Corollary 2. Let C be a nonzero negacyclic code of length 14 over Fpm . Then C = 〈(x2 + 1)i〉 ⊆ R1 for i ∈ {0, 1, . . . , 7}, (a) C is dual containing if and only if and 0 ≤ i ≤ 7/2. (b) C is self-orthogonal if and only if ps/2 ≤ i ≤ 7. (c) C is self-orthogonal do not exist. If pm ≡ 1 (mod 4), i.e., C = 〈(x− γ)i(x + γ)j 〉 ⊆ R1 for 0 ≤ i, j ≤ 7, (a) C is dual containing if and only if 0 ≤ i, j ≤ 7/2. (b) C is self-orthogonal if and only if 7/2 ≤ i, j ≤ 7. (c) C is self-dual if and only if i + j = 7. In the next section, we will use the concept of coefficient weight of polyno- mials, which was given in [19]: Let f(x) = anxn + an−1xn−1 + · · ·+ a1x + a0 be a polynomial with degree n, the coefficient weight of f , is defined as cw(f) = { 0, if f is a monomial min{|i− j| : ai = 0, aj = 0, i = j}, otherwise. Obviously, cw(f) is the smallest distance among exponents of nonzero terms of f(x). Base on this fact, we have the following lemma. Lemma 3. For any two nonzero polynomial f(x) and g(x), if f(x) and g(x) satisfied one of the following condition • 0 ≤ deg(g(x)) ≤ cw(f(x)) − 2, and deg(f(x)) + deg(g(x)) ≤ n− 2; • 0 ≤ deg(g(x)) = cw(f(x)) − 1, and deg(f(x)) + deg(g(x)) = n− 1. 22 A note on the Symbol-Pair Distance of... Then, by definition, wtsp(f(x) g(x)) = wtH(f(x)) · wtsp(g(x)). The condition deg(f(x)) + deg(g(x)) ≤ n − 2 ensures that f(x)g(x) does not represent a codeword of length n that has the first and last entries being nonzero, otherwise, wtH(f(x))·wtsp(g(x)) may be greater than wtsp(f(x) g(x)). The conditions 0 ≤ deg(g(x)) = cw(f(x))−1, and deg(f(x))+deg(g(x)) = n−1 ensures that a codeword of length n can be partition into some same short codes, otherwise, wtH(f(x)) ·wtsp(g(x)) may be less than wtsp(f(x) g(x)). This can be explained by the following two examples. Example 4. Let f(x) = x6 + 1, and g(x) = x4 + x2 + 1. Then cw(f(x)) = 6, deg(g(x)) = 4, wtH(f(x)) = 2, wtsp(g(x)) = 6, and f(x)g(x) = x10+x8+x6+ x4+x2+1. If the code of length is n = 11, then f(x)g(x) represents the codeword (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1), and wtsp(f(x)g(x)) = 11 < wtH(f(x)) · wtsp(g(x)). Example 5. Let f(x) = x6 + 1, and g(x) = x5 + x2 + 1. Then cw(f(x)) = 6, deg(g(x)) = 5, wtH(f(x)) = 2, wtsp(g(x)) = 5, and f(x)g(x) = x11 + x8 + x6 + x5 + x2 + 1. If the code of length is n = 13, then f(x)g(x) represents the codeword (1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0), and wtsp(f(x)g(x)) = 11 > wtH(f(x))· wtsp(g(x)). 3. Symbol-pair distance of negacyclic codes of length 14 over F7m As discussed in Section 3, for the sake of narrative convenience, we denote negacyclic code 〈(x2+1)i〉 by Ci for i = 0, 1, . . . , ps, and 〈(x+γ)i(x−γ)j 〉 by Ci,j for i, j ∈ {0, 1, . . . , 7}. Their symbol-pair distance are denoted by dsp(Ci) and dsp(Ci,j), respectively. For any codeword c(x) ∈ Ci or Ci,j, the Hamming weight and symbol-pair weight are denoted by wtH(c(x)) and wtsp(c(x)), respectively. In [3], the author have considered the Hamming distance of Ci. Proposition 6. (Theorem 7.9 of [3]) The negacyclic codes of length 14 over F7m are of the form Ci = 〈(x2+1)i〉 for i = 0, 1, · · · , ps. Moreover, its Hamming distance dH(Ci) is determined by: dH(Ci) = ⎧⎪⎪⎨ ⎪⎪⎩ 1, if i = 0 (β + 1)pk1 , if 7− 71−k1 + β7−k1 + 1 ≤ i ≤ 7− 71−k1 + (β + 1)7−k1 where 0 ≤ β ≤ 7, and 0 ≤ k1 0, if i = 7. N. T. Bac 23 Base on the Hamming distance, we can show the symbol-pair distance of Ci as follows. Theorem 7. The symbols defined as Proposition 6. Then symbol-pair distance of Ci is determined by: dsp(Ci) = ⎧⎪⎪⎨ ⎪⎪⎩ 2, if i = 0 2(β + 1)7k1, if 7− 71−k1 + β7−k1 + 1 ≤ i ≤ 7− 71−k1 + (β + 1)7−k1 where 0 ≤ β ≤ 5, and 0 ≤ k1 0, if i = 7. Proof. Recall that R1 = C0 ⊃ C1 ⊃ · · · ⊃ C6 ⊃ C7 = 〈0〉. Clearly, dsp(C7) = 0, and dsp(C0) = 2. Furthermore, 2 = dsp(C0) ≤ dsp(C1) ≤ dsp(C2) ≤ · · · ≤ dsp(C6). Now, we consider the other cases. Let c(x) be an arbitrary nonzero element of Ci. Then there exist a nonzero element f(x) ∈ R1 such that c(x) = (x2+1)if(x). By the Division Algorithm, we can assume that deg(f) < 14− 2i− 1. Let f(x) be expressed as f(x) = f0 + f1x + · · ·+ fx, where f0, f1, . . . , f ∈ Fpm , and  = 14 − 2i − 1. Partition f(x) into two polynomials f0(x) and f1(x), i.e., f(x) = f0(x) + f1(x), where f0(x) only contains terms of even exponents f2lx2l, and f1(x) only contains terms of odd exponents f2l+1x2l+1 , where l = 0, 1, · · · , 6− i. We consider the following 3 cases. Case 1: f0(x) = 0. Then there are exactly coefficients f2j+1 is nonzero. We have wtH(c(x)) = wtH((x2 + 1)if(x)) = wtH ([ i∑ h=0 ( i h ) x2h ] f1(x) ) . Thus, the nonzero terms of c(x) are 2h + 1 positions apart for h = 0, 1, · · · , ps − 1. It follows that wtsp(c(x)) = 2wtH(c(x)) ≥ 2 dH(Ci). Case 2: f1(x) = 0. Then there are exactly coefficients f2j is nonzero. Then wtH(c(x)) = wtH((x2 + 1)if(x)) = wtH ([ i∑ h=0 ( i h ) x2h ] f0(x) ) . So the nonzero terms of c(x) are 2h positions apart for h = 0, 1, · · · , ps−1. Therefore, wtsp(c(x)) = 2wtH(c(x)) ≥ 2 dH(Ci). 24 A note on the Symbol-Pair Distance of... Case 3: f0(x) = 0 and f1(x) = 0. Then (x2 + 1)if(x) =(x2 + 1)if0(x) + (x2 + 1)if1(x). Because the nonzero terms of (x2 + 1)if0(x) are 2k positions apart and the nonzero terms of (x2 + γ)if1(x) are 2k + 1 positions apart for k = 0, 1, · · · , ps−1, then (x2+1)if0(x) and (x2+1)if1(x) do not contain any term with same power of x. Therefore, wtH((x2 + 1)if(x)) = wtH((x2 + 1)if0(x)) + wtH((x2 + 1)if1(x)). Since (x2 + 1)if0(x) and (x2 + 1)if1(x) are nonzero element in Ci, wtH((x2 + 1)if0(x)) ≥ dH(Ci) and wtH((x2 + 1)if1(x)) ≥ dH(Ci). Hence, wtsp(c(x)) ≥ wtH((x2 + 1)if(x)) ≥ 2 dH(Ci). Theorefore, for any c(x) ∈ Ci, wtsp(c(x)) ≥ 2 dH(Ci), implying dsp(Ci) ≥ 2 dH(Ci). As dsp(Ci) ≤ 2 dH(Ci), making use of Proposition 6, the result follows.  Now, we consider the symbol-pair distance of negacyclic code Ci,j for i, j ∈ {0, 1, . . . , ps}. Obviously, if i = j = 0, then C0,0 = R1, and if i = j = ps, then Cps,ps = {0}. For the remaining values of i, j, as the symmetries of all the cases, without loss of generality, in the following of this section, we always assume i ≥ j. If i = j, clearly, Ci,j = 〈(x + γ)i(x− γ)j 〉 = 〈(x2 + 1)i〉. In fact, Theorem 7 gives the symbol-pair distance of Ci,j. Proposition 8. If i = j for i ∈ {0, 1, · · · , 7}. Then symbol-pair distance of Ci,j is determined by: dsp(Ci,j) = ⎧⎪⎪⎨ ⎪⎪⎩ 2, if i = 0 2(β + 1)7k1 , if 7− 71−k1 + β7−k1 + 1 ≤ i ≤ 7− 71−k1 + (β + 1)7−k1 where 0 ≤ β ≤ 5, and 0 ≤ k1 0, if i = 7. 4. Symbol-Pair negacyclic Codes of Length 14 over F7m In [15], the authors introduced that the parameters of an [n, k, dsp] linear code C over F7m satisfying dsp ≤ n−k+2, and if the equality hold then a symbol-pair N. T. Bac 25 code C is called an optimal code with respect to Singleton bound, or a maximum distance separable (MDS) symbol-pair code. For any fixed symbol-pair code length n and dimension k, MDS symbol-pair code has the largest minimum distance, i.e., they have the best possible error-correction capability. Hence, constructing MDS symbol-pair codes is significance in theory and practice. In this section, we will determine all MDS symbol-pair negacyclic codes of length 14. First of all, we consider the case pm ≡ 3 (mod 4). Theorem 9. Let p be an odd prime, m be an interger, pm ≡ 3 (mod 4), then the negacyclic codes of length 14 over F7m are of the form Ci = 〈(x2 + 1)i〉 for i = 0, 1, · · · , 7. Moreover, Ci is a MDS symbol-pair code if and only if one of the following conditions holds: • If s = 1, then i = β + 1, for 0 ≤ β ≤ 7, in such case, dsp(Ci) = 2(β + 2). Proof. For 0 ≤ i, j ≤ 7, by Proposition 1, we have |Ci| = pm(7−i−j), implying the dimension of symbol-pair code Ci is 7− i− j. By Singleton bound, Ci is a MDS symbol-pair code if and only if i + j = dsp(Ci)− 2. 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