A note on cyclic codes from APN functions

Abstract

Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding (SIAM J Discret Math 27(4):1977–1994, 2013), Ding and Zhou (Discret Math, 2014) constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. In this paper, we consider two open problems involving the inverse APN function \(f(x)=x^{q^m-2}\) 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 \(s_t={\mathrm {Tr}}((1+\alpha ^t)^e)\), where \(\alpha \) is a primitive element in \({\mathrm {GF}}(q)\). These techniques can also be applied to other open problems in Ding (SIAM J Discret Math 27(4):1977–1994, 2013), Ding and Zhou (Discret Math, 2014).