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Volume 38, Issue 2
Subfield Codes of Linear Codes from Perfect Nonlinear Functions and Their Duals

Dabin Zheng, Xiaoqiang Wang, Yayao Li & Mu Yuan

Commun. Math. Res., 38 (2022), pp. 157-183.

Published online: 2022-02

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  • 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 $p\geq 5$.

  • AMS Subject Headings

94B05, 94B25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-38-157, author = {Zheng , DabinWang , XiaoqiangLi , Yayao and Yuan , Mu}, title = {Subfield Codes of Linear Codes from Perfect Nonlinear Functions and Their Duals}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {38}, number = {2}, pages = {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 $p\geq 5$.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0520}, url = {http://global-sci.org/intro/article_detail/cmr/20269.html} }
TY - JOUR T1 - Subfield Codes of Linear Codes from Perfect Nonlinear Functions and Their Duals AU - Zheng , Dabin AU - Wang , Xiaoqiang AU - Li , Yayao AU - Yuan , Mu JO - Communications in Mathematical Research VL - 2 SP - 157 EP - 183 PY - 2022 DA - 2022/02 SN - 38 DO - http://doi.org/10.4208/cmr.2020-0520 UR - https://global-sci.org/intro/article_detail/cmr/20269.html KW - Subfield code, perfect nonlinear function, quadratic form, weight distribution, Sphere Packing bound. AB -

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 $p\geq 5$.

Dabin Zheng, Xiaoqiang Wang, Yayao Li & Mu Yuan. (2022). Subfield Codes of Linear Codes from Perfect Nonlinear Functions and Their Duals. Communications in Mathematical Research . 38 (2). 157-183. doi:10.4208/cmr.2020-0520
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