Binary primitive LCD BCH codes
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Binary primitive LCD BCH codes Xinmei Huang1,2
· Qin Yue2,3 · Yansheng Wu4 · Xiaoping Shi5 · Jerod Michel6
Received: 7 April 2020 / Revised: 10 August 2020 / Accepted: 19 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Linear complementary dual (LCD) codes have attracted much attention in recent years due to their applications in implementations against side-channel attacks and fault injection attacks. Comparing coset leaders, we introduce the new concept of absolute coset leaders, which provides advantages for constructing LCD BCH codes. We then give explicit presentations for the largest, second largest and third largest absolute coset leaders and use them to construct binary LCD BCH codes. Lastly, we determine the parameters of these LCD BCH codes. Keywords LCD codes · BCH codes · Absolute coset leaders Mathematics Subject Classification 94B05 · 94B15
1 Introduction Let Fq be a finite field with q elements, where q is a prime power. An [n, k, d] linear code C over Fq is a linear subspace of Fqn with dimension k and minimum (Hamming) distance d. Let Ai denote the number of codewords in C with Hamming weight i. The weight enumerator of the code C is characterized by the polynomial 1 + A1 z + A2 z 2 + · · · + An z n . The sequence (1, A1 , A2 , . . . , An ) represents the weight distribution of C . A code C is t-weight if the number of nonzero Ai ’s in the sequence (A1 , A2 , . . . , An ) is equal to t. The Euclidean inner product of two vectors a = (a0 , . . . , an−1 ), c = (c0 , . . . , cn−1 ) ∈ Fqn n−1 is denoted by a, c = ac T = i=0 ai ci . Then the dual code of a typical [n, k] linear code C is defined as: C ⊥ = {a ∈ Fqn : ac T = 0 for all c ∈ C }.
Next, we consider a special class of linear codes called cyclic codes. If the code C satisfies the condition that for each codeword (c0 , c1 , . . . , cn−1 ) ∈ C , the codeword (cn−1 , c0 , c1 , . . . , cn−2 ) is also in C , then C is said to be a cyclic code. A cyclic code C of length n over Fq corresponds to an ideal of the quotient ring Fq [x]/x n − 1. Furthermore,
Communicated by C. Ding.
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Xinmei Huang [email protected]
Extended author information available on the last page of the article
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X. Huang et al.
Fq [x]/x n − 1 is a principle ideal ring and C is generated by a monic divisor g(x) of x n − 1. Here g(x) is called the generator polynomial of the code C , and we write C = g(x). Let Zn = {0, 1, . . . , n − 1} be the ring of integers modulo n. For s ∈ Zn , assume that ls is the smallest positive integer such that sq ls ≡ s (mod n). Then the q-cyclotomic coset containing s is defined as Cs = {s, sq, . . . , sq ls −1 }
(mod n) ⊂ Zn .
The smallest integer in Cs is called the coset leader of Cs (see [11]). In this paper, for the purpose of investigating LCD BCH codes, we define the absolute coset leader of Cs to be the smallest positive integer in the set {k, n − k : k ∈ Cs }. Let m = ordn (q) be the multiplicative order of q modulo n and γ a primitive element of q m −1
Fq m . Then α = γ
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