A Note on Cyclic Codes from APN Functions

TL;DR

This paper investigates cyclic codes derived from specific APN functions, determining their dimensions and minimum weight bounds, and provides a general framework for analyzing related sequences in coding theory.

Contribution

It addresses open problems by analyzing cyclic codes from inverse and Dobbertin APN functions, offering new methods for minimal polynomial and linear span calculations.

Findings

Dimensions of cyclic codes from inverse and Dobbertin APN functions determined.

Lower bounds on minimum weight of these cyclic codes established.

A general framework for analyzing sequences related to these codes is presented.

Abstract

Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding in \cite{D} constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented ten open problems on cyclic codes from highly nonlinear functions. In this paper, we consider two open problems involving the inverse APN functions $f(x)=x^{q^m-2}$ and the Dobbertin APN function $f(x)=x^{2^{4i}+2^{3i}+2^{2i}+2^{i}-1}$. From the calculation of linear spans and the minimal polynomials of two sequences generated by these two classes of APN functions, the dimensions of the corresponding cyclic codes are determined and lower bounds on the minimum weight of these cyclic codes are presented. Actually, we present a framework for the minimal polynomial and linear span of the sequence $s^{\infty}$ defined by $s_t=Tr((1+\alpha^t)^e)$, where $\alpha$ is a primitive element in $GF(q)$. These techniques can also be applied into other open problems in \cite{D}.