Investigation of the permutation and linear codes from the Welch APN function

Abstract

Dobbertin in 1999 proved that the Welch power function \(x^{2^m+3}\) was almost perferct nonlinear (APN) over the finite field \(\mathbb {F}_{2^{2m+1}}\), where m is a positive integer. In his proof, Dobbertin showed that the APNness of \(x^{2^m+3}\) essentially relied on the bijectivity of the polynomial \(g(x)=x^{2^{m+1}+1}+x^3+x\) over \(\mathbb {F}_{2^{2m+1}}\). In this paper, we first determine the differential and Walsh spectra of the permutation polynomial g(x), revealing its favourable cryptograhphic properties. We then explore four families of binary linear codes related to the Welch APN power functions. For two cyclic codes among them, we propose algebraic decoding algorithms that significantly outperform existing methods in terms of decoding complexity.