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The subfield codes of several classes of linear codes

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Abstract

Let \(\mathbb {F}_{2^{m}}\) 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

$$ \mathcal{C}_{a,b}=\left\{((\text{Tr}_{1}^{m}(a f(x)+bx)+c)_{x \in \mathbb{F}_{2^{m}}}, \text{Tr}_{1}^{m}(a), \text{Tr}_{1}^{m}(b)) : a,b \in \mathbb{F}_{2^{m}}, c \in \mathbb{F}_{2}\right\}, $$

for f(x) = x2 and f(x) = x6 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 of \(\mathcal {C}_{a,b}\) for \(f(x)=x^{2^{i}+2^{j}}\), 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 \(\mathcal {C}_{a,b}\) as follows:

$$ \mathcal{C}_{a}=\left\{((\text{Tr}_{1}^{m}(a x^{2^{i}+2^{j}}+bx)+c)_{x \in \mathbb{F}_{2^{m}}}, \text{Tr}_{1}^{m}(a)) : a,b \in \mathbb{F}_{2^{m}}, c \in \mathbb{F}_{2}\right\}, $$

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.

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Acknowledgments

The authors sincerely thank the reviewers and the editor for their helpful comments and valuable suggestions, which have improved the presentation of this paper. This work was partially supported by National Natural Science Foundation of China under Grant 11971156 and Hubei province science and technology innovation major project under Grant 2019ACA144.

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  1. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

    Xiaoqiang Wang & Dabin Zheng

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  1. Xiaoqiang Wang
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Correspondence to Dabin Zheng.

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Wang, X., Zheng, D. The subfield codes of several classes of linear codes. Cryptogr. Commun. 12, 1111–1131 (2020). https://doi.org/10.1007/s12095-020-00432-4

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  • DOI: https://doi.org/10.1007/s12095-020-00432-4

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