Two families of subfield codes with a few weights
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Two families of subfield codes with a few weights Can Xiang1 · Wenjuan Yin2 Received: 19 June 2020 / Accepted: 22 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, two families of binary subfield codes with a few weights are presented from two special classes of linear codes, and their parameters are explicitly determined. Moreover, the parameters of the duals of these subfield codes are also studied. The two infinite families of subfield codes presented in this paper are distanceoptimal with respect to the Griesmer bound and their duals are almost distance-optimal with respect to the sphere-packing bound. Keywords Linear code · Subfield code · Distance-optimal Mathematics Subject Classification (2010) 94A24 · 94B35 · 94B15 · 94A55
1 Introduction Let q be a prime power and GF(q) be a finite field with q elements. An [n, k, d] code C over GF(q) is a k-dimensional subspace of GF(q)n with length n, dimension k and minimum (Hamming) distance d, where n, k, d are positive integers. Denote by C ⊥ the dual code of a linear code C . Let Ai denote the number of codewords with Hamming weight i in a code C of length n. The weight enumerator of C is defined by 1 + A1 z + A2 z2 + · · · + An zn . The weight distribution (1, A1 , . . . , An ) is an important research topic in coding theory, as it contains crucial information about the error correcting capability of the code [20]. Thus the study of the weight distribution attracts much attention in coding theory and much work focuses on the determination of the weight distributions of linear codes (see, for example, [5–10, 15, 18, 19, 22–25, 27, 28]). Can Xiang
[email protected] Wenjuan Yin [email protected] 1
College of Mathematics and Informatics, South China Agricultural University, Guangzhou, Guangdong 510642, China
2
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
Cryptography and Communications
A code C is said to be a t-weight code if the number of nonzero Ai in the sequence (A1 , A2 , · · · , An ) is equal to t. Let fi (x1 , x2 , · · · , xs )(1 ≤ i ≤ t) be an s-variable polynomial over GF(q). Denote ⎡
Gx1 ,x2 ,··· ,xs
⎤ f1 (x1 , x2 , · · · , xs ) ⎢f2 (x1 , x2 , · · · , xs )⎥ ⎥ =⎢ ⎣ ⎦ ··· ft (x1 , x2 , · · · , xs ) (x
s 1 ,x2 ,··· ,xs )∈GF(q)
which is a t × q s matrix over GF(q). s We define a linear code CV of length q + u over GF(q) with generator matrix GV = Gx1 ,x2 ,··· ,xs A , where 0 ≤ u ≤ t and A is a t × u matrix which consists of u distinct column vectors of the identity matrix It . This construction is generic in the sense that many classes of known codes could be produced by properly selecting the generator matrix GV . An [n, k, d] code over GF(q) is said to be distance-optimal if there is no [n, k, d ] code over GF(q) with d > d and
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