We compute the weight distribution of binary cyclic codes with arbitrarily many generalized Niho-type nonzeroes, where the exponents of the nonzeroes nearly form a long arithmetic progression. Determining their weight distribution requires two distinct techniques. The first involves using the polar representation associated with generalized Niho-type exponents and leveraging the subtle structure of the nonzeroes whose exponents are close to an arithmetic progression. The second applies the weight distribution of binary Zetterberg codes to address certain counting problems. This resolves two open questions raised by Li and Zeng.

The favorable algebraic properties and error-correcting performance of BCH codes have motivated extensive research and diverse applications, whereas the study of extended BCH codes remains relatively less explored. In this paper, we focus on the extended BCH codeC(q,q+1,δ,h)over the finite field Fq. Specifically, we investigate the parameters of three families of extended BCH codes and the weight distributions of their duals by solving certain quadratic and quartic equations:C(q,q+1,3,1)is shown to be NMDS when gcd(q+1,3) = 1;C(q,q+1,δ,q)with 3 ≤ δ ≤ ⌈(q+ 5)/2⌉ is proved to be AMDS; andC(q,q+1,3,q/2)is MDS for evenq.

The cross-correlation problem is a classic problem in sequence design. In this paper we compute the cross-correlation distribution of the Niho-type decimation$d=3(p^{m}-1)+1$over${\mathrm {GF}}(p^{2m})$for any prime$p \ge 5$. Previously this problem was solved by Xia et al. only for$p=2$and$p=3$in a series of papers. The main difficulty of this problem for$p \ge 5$, as pointed out by Xia et al., is to count the number of codewords of “pure weight” 5 in p-ary Zetterberg codes. It turns out this counting problem can be transformed by the MacWilliams identity into counting codewords of weight at most 5 in p-ary Melas codes, the most difficult of which is related to a K3 surface well studied in the literature and can be computed. When$p \ge 7$, the theory of elliptic curves over finite fields also plays an important role in the resolution of this problem.

Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction framework of 3-dimensional linear codesCf,gover Fqmparameterized by any two functionsf,gover Fqm, and then study the properties of six types ofCf,g, its punctured codeC*f,gand their corresponding subfield codes over Fq. The classification ofCf,gis based on special choices off,gas trace function, norm function, almost bent function, Boolean bent function or a combination of these functions. For the first two types ofCf,g, we explicitly determine the weight distributions and dualities ofCf,g, C*f,gand their subfield codes over Fq. The remaining four types ofCf,gare restricted toq= 2, and the weight distributions and dualities of the subfields codeC(q)f,gandC*f,g(q)are completely determined. Most of the resultant linear codes (over Fqmor over Fq) have few weights. Some of them are optimal and some have the best-known parameters according to the tables maintained at http://www.codetables.de. In fact, 16 infinite families of optimal linear codes are produced in this paper. As a byproduct, a family of [24m-2, 2m+ 1, 24m-3] quaternary Hermitian self-orthogonal codes are obtained withm≥ 2. As an application, we present several infinite families of 2-designs or 3-designs with some of the codes presented in this paper.

Permutation polynomials over finite fields are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. This family of polynomials constitutes an active research area in which advances are being made constantly. In particular, constructing infinite classes of permutation polynomials over finite fields with good differential properties (namely, low) remains an exciting problem despite much research in this direction for many years. This article exhibits low differentially uniform power permutations over finite fields of odd characteristics. Specifically, its objective is twofold concerning the power functions$F(x)=x^{\frac {p^{n}+3}{2}}$defined over the finite field${\mathbb {F}}_{p^{n}}$of order$p^{n}$, where$p$is an odd prime, and$n$is a positive integer. The first is to complement some former results initiated by Helleseth and Sandberg in 1997 by solving the open problem left open for more than twenty years concerning the determination of the differential spectrum of$F$when$p^{n}\equiv 3\pmod 4$and$p\neq 3$. The second is to determine the exact value of its differential uniformity. Our achievements are obtained firstly by evaluating some character sums over${\mathbb {F}}_{p^{n}}$(which amounts to evaluating the number of${\mathbb {F}}_{p^{n}}$-rational points on some related curves and secondly by computing the number of solutions in$({\mathbb {F}}_{p^{n}})^{4}$of a system of equations presented by Helleseth, Rong, and Sandberg, naturally appears while determining the differential spectrum of$F$. We show that in the considered case ($p^{n}\equiv 3\pmod 4$and$p\neq 3$),$F$is an APN power permutation when$p^{n}=11$, and a differentially 4-uniform power permutation otherwise.

Let$n$be a positive integer and$p$a prime. The power mapping$x^{p^{n}-3}$over${\mathbb {F}}_{p^{n}}$has desirable differential properties, and its differential spectra for$p=2,\,3$have been determined. In this paper, for any odd prime$p$, by investigating certain quadratic character sums and some equations over${\mathbb {F}}_{p^{n}}$, we determine the differential spectrum of$x^{p^{n}-3}$with a unified approach. The obtained result shows that for any given odd prime$p$, the differential spectrum can be expressed explicitly in terms of$n$. Compared with previous results, a special elliptic curve over${\mathbb {F}}_{p}$plays an important role in our computation for the general case$p \ge 5$.

In a prior paper (Ellingsenet al., 2020), two of us, along with P. Ellingsen, P. Felke, and A. Tkachenko, defined a new (output) multiplicative differential and the corresponding$c$-differential uniformity, which has the potential of extending differential cryptanalysis. Here, we continue the work by looking at some APN functions through the mentioned concept and showing that their$c$-differential uniformity increases significantly in some cases.

Functions with low differential uniformity have important applications in cryptography, coding theory, and sequence design. The differential spectrum of a cryptographic function is of great interest for estimating its resistance to some variants of differential cryptanalysis. Finding power permutations (i.e., monomial bijective mappings) over finite fields with low differential uniformity and determining their differential spectra have received a lot of attention over the past two decades. The objective of this paper is to study the differential properties of the well-known Kasami power permutations x p2k-pk+1 over GF(pn), where p is an odd prime and k is an integer with gcd(n, k) = 1. It turns out that this family of monomials is differentially (p + 1)-uniform. Our result in the case of p = 3 gives an affirmative solution to a recent conjecture by Xu, Cao, and Xu. Most notably, the differential spectrum of this family of power permutations is completely determined.