The subfield codes of several classes of linear codes
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The subfield codes of several classes of linear codes Xiaoqiang Wang1 · Dabin Zheng1 Received: 2 December 2019 / Accepted: 19 March 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let F2m be the finite field with 2m elements, where m is a positive integer. Recently, Heng and Ding in (Finite Fields Appl. 56:308–331, 2019) studied the subfield codes of two families of hyperovel codes and determined the weight distribution of the linear code m m Ca,b = ((Trm 1 (af (x) + bx) + c)x∈F2m , Tr1 (a), Tr1 (b)) : a, b ∈ F2m , c ∈ F2 , for f (x) = x 2 and f (x) = x 6 with odd m. Let v2 (·) denote the 2-adic order function. This paper investigates more subfield codes of linear codes and obtains the weight distribution i j of Ca,b for f (x) = x 2 +2 , where i, j are nonnegative integers such that v2 (m) ≤ v2 (i − j ) (i ≥ j ). In addition to this, we further investigate the punctured code of Ca,b as follows: 2i +2j m m , c ∈ F2 , Ca = ((Trm (ax + bx) + c) , Tr (a)) : a, b ∈ F m 2 x∈F 1 1 2 and determine its weight distribution for any nonnegative integers i, j . The parameters of these binary linear codes are new in most cases. Some of the codes and their duals obtained are optimal or almost optimal. Keywords Weight distribution · Subfield code · Linear code · Optimal code Mathematics Subject Classification (2010) 94B05 · 94B15
1 Introduction Linear codes over finite fields have been studied extensively because of their linear structures and practical implementations. It is the basis of the research of various kinds of codes. Many well-known codes, such as BCH, Golay, Reed-Muller, Preparata, Goppa, and Hamming codes, are special cases of linear codes.
Dabin Zheng
[email protected] Xiaoqiang Wang [email protected] 1
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China
Cryptography and Communications
Let q be a prime power and n be a positive integer. An [n, k, d] code C over the finite field Fq is a k-dimensional linear subspace of Fnq with minimum Hamming distance d. The Hamming weight of a codeword c = (c0 , c1 , · · · , cn−1 ) ∈ C , which is denoted by wtH (c), is the number of nonzero entry ci , where 0 ≤ i ≤ n − 1. For a linear code C of length n over Fq , the minimum Hamming distance of C is given by: dH (C ) = min{wtH (c) | c = 0, c ∈ C }. Let Ai denote the number of nonzero codewords with Hamming weight i in C . The weight enumerator of C is defined by 1 + A1 x + A2 x 2 + · · · + An x n . The weight distribution of C is defined by the sequence (1, A1 , · · · , An ). The weight distribution of a code not only gives the error correcting ability of the code, but also allows the computation of the error probability of error detection and correction [20]. Hence, the study of the weight distribution of a linear code is important in both theory and applications. In recent years, the weight distributions of linear codes or cyclic codes have been intensively studied. In [13], the authors presented the recent pr
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