@Article{CMR-38-2,
author = {Zheng, Dabin and Xiaoqiang, Wang and Yayao, Li 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 = {<p style="text-align: justify;">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 $\mathbb{F}_{p^m}$&nbsp;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 $p\geq 5$.</p>},
issn = {2707-8523},
doi = {https://doi.org/10.4208/cmr.2020-0520},
url = {https://global-sci.com/article/81482/subfield-codes-of-linear-codes-from-perfect-nonlinear-functions-and-their-duals}
}