Year: 2022
Author: Dabin Zheng, Xiaoqiang Wang, Yayao Li, Mu Yuan
Communications in Mathematical Research , Vol. 38 (2022), Iss. 2 : pp. 157–183
Abstract
Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is an odd prime and $m$ is a positive integer. Recently, [17] and [35] determined the weight distributions of subfield codes with the form $$\mathcal{C}_f=\{(({\rm Tr}(af(x)+bx)+c)_{x\in \mathbb{F}_{p^m}},{\rm Tr}(a)):a,b\in \mathbb{F}_{p^m},c\in \mathbb{F}_p\}$$ for $f(x) = x^2$ and $f(x) = x^{p^k+1}$ , respectively, where $Tr(·)$ is the trace function from $\mathbb{F}_{p^m}$ to $\mathbb{F}_p$, and $k$ is a nonnegative integer. In this paper, we further investigate the subfield code $\mathcal{C}_f$ for $f(x)$ being a known perfect nonlinear function over $\mathbb{F}_{p^m}$ and generalize some results in [17, 35]. The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields. In addition, the parameters of the duals of these codes are also determined. Several examples show that some of our codes and their duals have the best known parameters according to the code tables in [16]. The duals of some proposed codes are optimal according to the Sphere Packing bound if $p\geq 5$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cmr.2020-0520
Communications in Mathematical Research , Vol. 38 (2022), Iss. 2 : pp. 157–183
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 27
Keywords: Subfield code perfect nonlinear function quadratic form weight distribution Sphere Packing bound.