Search

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]

Combinatorial t‐designs have nice applications in coding theory, finite geometries, and several engineering areas. A classical method for constructing t‐designs is by the action of a permutation group that is t‐transitive or t‐homogeneous on a point set. This approach produces t‐designs, but may not yield (t+1)‐designs. The objective of this paper is to study how to obtain 3‐designs with 2‐transitive permutation groups. The incidence structure formed by the orbits of a base block under the action of the general affine groups, which are 2‐transitive, is considered. A characterization of such incidence structure to be a 3‐design is presented, and a sufficient condition for the stabilizer of a base block to be trivial is given. With these general results, infinite families of 3‐designs are constructed by employing almost perfect nonlinear functions. Some 3‐designs presented in this paper give rise to self‐dual binary codes or linear codes with optimal or best parameters known. Several conjectures on 3‐designs and binary codes are also presented. [ABSTRACT FROM AUTHOR]

The investigation of partially APN functions has attracted a lot of research interest recently. In this paper, we present several new infinite classes of 0-APN power functions over F2n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {F}_{2^n}$$\end{document} by using the multivariate method and resultant elimination, and show that these 0-APN power functions are CCZ-inequivalent to the known ones. [ABSTRACT FROM AUTHOR]

Differentially 4-uniform permutations onwith high nonlinearity are often chosen as substitution boxes in block ciphers. Recently, Quet al.used the powerful switching method to construct permutations with low differential uniformity from the inverse function [Quet al., Constructing differentially 4-uniform permutations over via the switching method, IEEE Trans. Inform. Theory 59(7) (2013), pp. 4675–4686, Quet al., More constructions of differentially 4-uniform permutations on, Des. Codes Cryptogr. DOI 10.1007/s10623-014-0006-x] and proposed a sufficient but not necessary condition for these permutations to be differentially 4-uniform. In this paper, a sufficient and necessary condition is presented. We also give a compact estimation for the number of constructed differentially 4-uniform permutations. Comparing with those constructions in [11,12], the number of functions constructed here is much bigger. As an application, a new class of differentially 4-uniform permutations is constructed. The obtained functions in this paper may provide more choices for the design of substitution boxes. [ABSTRACT FROM AUTHOR]

In this paper, we propose three new classes of permutation trinomials over F q 3 . The first two classes are of the form x + L (x q 2 + q − 1) , where L (x) ∈ { x + A x q , x + A x q 2 } , q even, and the third class is x + A x q 2 − q + 1 + A 2 x q 2 , q odd. In fact we prove a conjecture proposed by Gong, Gao and Liu [8] about permutation trinomials of the form x q + 1 + L (x) ∈ F q 3 [ x ] as a particular case of our results. Moreover, we determine necessary and sufficient conditions for the permutation trinomials studied. We also show that the permutation trinomials proposed in this paper are new in the sense that they are not quasi-multiplicative equivalent to permutation trinomials over F q 3 known so far. [ABSTRACT FROM AUTHOR]

Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However, APN functions on finite fields of odd characteristic do not satisfy desired properties. In this paper, we modify the definition of APN function in the case of odd characteristic, and study its properties. [ABSTRACT FROM AUTHOR]

This article determines all the solutions in the finite field F 2 4 n of the equation x 2 3 n + 2 2 n + 2 n − 1 + (x + 1) 2 3 n + 2 2 n + 2 n − 1 = b. Specifically, we explicitly determine the set of b 's for which the equation has i solutions for any positive integer i. Such sets, which depend on the number of solutions i , are given explicitly and expressed nicely, employing the absolute trace function over F 2 n , the norm function over F 2 4 n relatively to F 2 n and the set of (2 n + 1) st roots of unity in F 2 4 n . The equation considered in this paper comes from an article by Budaghyan et al. ([2]) in which the authors have investigated novel approaches for obtaining alternative representations for functions from the known infinite APN families. In particular, they have been interested in determining the differential spectrum of some power functions among them is the one F (x) = x 2 3 n + 2 2 n + 2 n − 1 defined over F 2 4 n . The problem of the determination of such spectrum has led to a conjecture (Conjecture 27 in the preprint (2020) [2] for which an updated version will appear in 2022 at the IEEE Transactions Information Theory) stated by Budaghyan et al. As an immediate consequence of our results, we prove that the above equation has 2 2 n solutions for one value of b , 2 2 n − 2 n solutions for 2 n values of b in F 2 4 n and has at most two solutions for all remaining points b , leading to complete proof of the conjecture raised by Budaghyan et al. We highlight that the recent work of Li et al., in [9] gives the complete differential spectrum of F and also gives an affirmative answer to the conjecture of Budaghyan et al. However, we emphasize that our approach is interesting and promising by being different from Li et al. Indeed, on the opposite to their article, our technique allows to determine ultimately the set of b 's for which the considered equation has solutions as well as the solutions of the equation for any b in F 2 4 n . [ABSTRACT FROM AUTHOR]

In this paper, we establish a lower bound on the total number of inequivalent APN functions on the finite field with 2 2 m elements, where m is even. We obtain this result by proving that the APN functions introduced by Pott and the second author [22] , which depend on three parameters k , s and α , are pairwise inequivalent for distinct choices of the parameters k and s. Moreover, we determine the automorphism group of these APN functions. [ABSTRACT FROM AUTHOR]