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Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$ -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$ -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes. [ABSTRACT FROM AUTHOR]

In this paper, we present two new families of APN functions. The first family is in bivariate form $\big (x^{3}+xy^{2}+ y^{3}+xy, x^{5}+x^{4}y+y^{5}+xy+x^{2}y^{2} \big)\,\,\vphantom {_{\int _{\int }}}$ over ${\mathbb F}_{2^{m}}^{2}$. It is obtained by adding certain terms of the form $\sum _{i}(a_{i}x^{2^{i}}y^{2^{i}},b_{i}x^{2^{i}}y^{2^{i}})$ to a family of APN functions recently proposed by Gölo&gcaron;lu. The $\vphantom {_{\int _{\int }}}$ second family has the form $L(z)^{2^{m}+1}+vz^{2^{m}+1}$ over ${\mathbb F}_{{2^{3m}}}$ , which generalizes a family of APN functions by Bracken et al. from 2011. By calculating the $\Gamma $ -rank of the constructed APN functions over ${\mathbb F}_{2^{8}}$ and ${\mathbb F}_{2^{9}}$ , we demonstrate that the two families are CCZ-inequivalent to all known families. In addition, the two new families cover two known sporadic APN instances over ${\mathbb F}_{2^{8}}$ and ${\mathbb F}_{2^{9}}$ , which were found by Edel and Pott in 2009 and by Beierle and Leander in 2021, respectively. [ABSTRACT FROM AUTHOR]

For every positive integers $n$ and $m$ and every even positive integer $\delta $ , we derive inequalities satisfied by the Walsh transforms of all vectorial $(n,m)$ -functions and prove that the case of equality characterizes differential $\delta $ -uniformity. This provides a generalization to all differentially $\delta $ -uniform functions of the characterization of Almost Perfect Nonlinear (APN) functions due to Chabaud and Vaudenay, by means of the fourth moment of the Walsh transform. Such generalization has been missing since the introduction of the notion of differential uniformity by Nyberg in 1994 and since Chabaud-Vaudenay’s result in the same year. Moreover, for each even $\delta \geq 2$ , we find several (in fact, an infinity of) such characterizations. In particular, when $\delta =2$ and $\delta =4$ , we have that, for any $(n,n)$ -function [resp. any $(n,n-1)$ -function)], the arithmetic mean of $W_{F}^{2}(u_{1},v_{1})W_{F}^{2}(u_{2},v_{2})W_{F}^{2}(u_{1}+u_{2},v_{1}+v_{2})$ when $u_{1},u_{2}$ range independently over ${\mathbb F}_{2}^{n}$ and $v_{1},v_{2}$ are nonzero and distinct and range independently over ${\mathbb F}_{2}^{m}$ is at least $2^{3n}$ , and that $F$ is APN (resp. is differentially 4-uniform) if and only if this arithmetic mean equals $2^{3n}$ (which is the value we would get with a bent function if such function could exist). These inequalities give more knowledge on the Walsh spectrum of $(n,m)$ -functions. We deduce in particular a property of the Walsh support of highly nonlinear functions. We also consider the completely open question of knowing if the nonlinearity of APN functions is necessarily non-weak (as it is the case for known APN functions); we prove new lower bounds which cover all power APN functions (and hence a large part of known APN functions), which explain why their nonlinearities are not bad, and we discuss the question of the nonlinearity of APN quadratic functions (since almost all other known APN functions are quadratic). [ABSTRACT FROM AUTHOR]