A class of subfield codes of linear codes and their duals
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A class of subfield codes of linear codes and their duals Xiaoqiang Wang1 · Dabin Zheng1 · Yan Zhang2 Received: 1 July 2020 / Accepted: 30 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Recently, subfield codes of some optimal linear codes have been studied. In this paper, we further investigate a class of subfield codes and generalize the results of the subfield codes of the conic codes in Ding and Wang (Finite Fields Appl. 56, 308–331, 2020). The weight distributions of these subfield codes and the parameters of their duals are determined. Some of the presented codes are optimal or almost optimal according to Grassl (2020) and their duals are distance-optimal with respect to the Sphere Packing bound if p > 3. As a byproduct, we directly obtain the weight distributions of the punctured codes, which is the same with the results presented in Du et al. (2019a, b), and determine the parameters of the duals of the punctured codes. These dual codes are distance-optimal with respect to the Sphere Packing bound with rare exceptions. Keywords Linear code · Subfield code · Weight distribution · Exponential sum · Sphere Packing bound Mathematics Subject Classification (2010) 94B05 · 94B15
1 Introduction Let p be an odd prime and Fpm be a finite field of size p m . An [n, k, d] code C over the finite field Fpm is a k-dimensional linear subspace of Fnpm with the minimum Hamming distance d. An [n, k, d] code is called distance-optimal if there does not exist [n, k, d + 1] code [7, 22]. The Hamming weight of a codeword c = (c0 , c1 , · · · , cn−1 ) ∈ C is the number of nonzero Dabin Zheng
[email protected] Xiaoqiang Wang [email protected] Yan Zhang [email protected] 1
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China
2
School of Computer Science and Information Engineering, Hubei University, Wuhan 430062, China
Cryptography and Communications
ci for 0 ≤ i ≤ n − 1. Let Ai denote the number of nonzero codewords with Hamming weight i in C . The weight enumerator of C is defined as 1 + A1 x + A2 x 2 + · · · + An x n . The sequence (1, A1 , · · · , An ) is called the weight distribution of C . 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 [25]. Hence, the study of the weight distribution of a linear code is important in both theory and applications. The reader can refer to [9–13] and the references therein. It is well known that the minimum Hamming distances of linear codes play an important role in measuring error-correcting performance. Thus, how to find optimal linear codes with new lengths and minimum distances is one of the central topics in coding theory. In recent years, a series of work have been done. Ding and Helleseth [8] presented some distance-optimal ternary cyclic codes of parameters [3m − 1, 3m − 2m − 1, 4] according to the Sphere Packing bound by utilizing
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