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Recently, a new concept called multiplicative differential cryptanalysis and the corresponding c -differential uniformity were introduced by Ellingsen et al. (2020), and then some low differential uniformity functions were constructed. In this paper, we further study the constructions of perfect c -nonlinear (PcN) power functions. First, we give a conjecture on all power functions to be PcN over GF (2 m). Second, several classes of PcN power functions are obtained over finite fields of odd characteristic for c = − 1 and our theorems generalize some results in Bartoli and Timpanella (2020), Hasan et al. (2021) and Zha and Hu (2021). Finally, the c -differential spectrum of a class of almost perfect c -nonlinear (APcN) power functions is determined. [ABSTRACT FROM AUTHOR]

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(ƒ) = {Tr(aƒ(x) + bx)x∈Fqm*: a, b ∈ Fqm}, where q is a prime power, Fqm* = Fqm \ {0}, Tr is the trace function from Fqm to Fq, and ƒ(x) is a function from Fqm to Fqm with ƒ(0) = 0. Almost bent functions, quadratic functions and some monomials on F2m were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper. [ABSTRACT FROM AUTHOR]

An $(r, \delta)$ -locally repairable code ($(r, \delta)$ -LRC for short) was introduced by Prakash et al. for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$ -LRCs produced by Gopalan et al.. An $(r, \delta)$ -LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. generalized the construction of cyclic $r$ -LRCs proposed by Tamo et al. , and constructed several classes of optimal $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of , , this paper first characterizes $(r, \delta)$ -locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively from the product of two sets of zeros. Our constructions include all optimal cyclic $(r,\delta)$ -LRCs proposed in , , and our method seems more convenient to obtain optimal cyclic $(r, \delta)$ -LRCs with flexible parameters. Moreover, many optimal cyclic $(r,\delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively with $(r+\delta -1)\nmid n$ can be obtained from our method. [ABSTRACT FROM AUTHOR]

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications, such as cryptography and coding theory. Following the idea by Wang to characterize the involutory behavior of the generalized cyclotomic mappings, this paper gives a more concise criterion for $x^{r}h(x^{s})\in {\mathbb F} _{q}[x]$ being involutions over the finite field ${\mathbb F}_{q}$ , where $r\geq 1$ and $s\,|\, (q-1)$. By using this criterion, we propose a general method to construct involutions of the form $x^{r}h(x^{s})$ over ${\mathbb F}_{q}$ from given involutions over some subgroups of ${\mathbb F}_{q}^{*}$ by solving congruent and linear equations over finite fields. Then, many classes of explicit involutions of the form $x^{r}h(x^{s})$ over ${\mathbb F}_{q}$ are obtained. [ABSTRACT FROM AUTHOR]

Abstract Let q be a power of a prime and F q be a finite field with q elements. In this paper, we propose four families of infinite classes of permutation trinomials having the form c x − x s + x q s , and investigate the relationship between two types of widely discussed permutation polynomials. Based on this relation, many classes of permutation polynomials having the form (x q k − x + δ) s + c x without restriction on δ are derived. [ABSTRACT FROM AUTHOR]

In this paper, several classes of two-weight or three-weight linear codes over Fp from quadratic or non-quadratic functions are constructed and their weight distributions are determined. From the constructed codes, we obtain some optimal linear codes with respect to the Singleton bound and the Griesmer bound. These two- or three-weight linear codes may have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. [ABSTRACT FROM AUTHOR]

This note presents two classes of permutation polynomials of the form $$(x^{p^m}-x+\delta )^s+L(x)$$ over the finite fields $${{\mathbb {F}}}_{p^{2m}}$$ as a supplement of the recent works of Zha, Hu and Li, Helleseth and Tang. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and m and k be two positive integers with m ≥ 3 . Let h ± 1 ( x ) and h ± t ( x ) be the minimal polynomials of ± α − 1 and ± α − t over F p , respectively, where α is a primitive element of F p m . Let C 1 , − 1 , ± t , C ± 1 , t , − t and C 1 , − 1 , t , − t be the cyclic codes over F p of length p m − 1 with parity-check polynomials h 1 ( x ) h − 1 ( x ) h ± t ( x ) , h ± 1 ( x ) h t ( x ) h − t ( x ) and h 1 ( x ) h − 1 ( x ) h t ( x ) h − t ( x ) , respectively. This paper determines the weight distributions of the cyclic codes C 1 , − 1 , ± t , C ± 1 , t , − t and C 1 , − 1 , t , − t for the parameter t satisfying some congruence equations. [ABSTRACT FROM AUTHOR]

For two positive integers m and k , let C e be a class of cyclic code of length 2 m − 1 over F 2 with three nonzeros γ − 1 , γ − ( 2 k + 1 ) and γ − ( 2 e k + 1 ) for e = 2 or 3, where γ is a primitive element of F 2 m . When m gcd ⁡ ( m , k ) is odd, Kasami in 1971 determined the weight distributions of cyclic codes C 2 and C 3 , which is the same as that of the dual of the triple error-correcting BCH code. This paper obtains the weight distributions of C 2 and C 3 for the case of m gcd ⁡ ( m , k ) being even. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and m , k be positive integers with m ≥ 3 k . Let C 1 and C 2 be cyclic codes over F p with parity-check polynomials h 2 ( x ) h 3 ( x ) and h 1 ( x ) h 2 ( x ) h 3 ( x ) , respectively, where h 1 ( x ) , h 2 ( x ) and h 3 ( x ) are the minimal polynomials of γ − 1 , γ − ( p k + 1 ) and γ − ( p 3 k + 1 ) over F p , respectively, for a primitive element γ of F p m . Recently, Zeng et al. (2010) obtained the weight distribution of C 2 for m gcd ( m , k ) being odd. In this paper, we determine the weight distribution of C 1 , and the weight distribution of C 2 for the case that m gcd ( m , k ) is even. [ABSTRACT FROM AUTHOR]

We give a necessary and sufficient condition such that the class of $$p$$ p -ary binomial functions proposed by Jia et al. (IEEE Trans Inf Theory 58(9):6054–6063, 2012 ) are regular bent functions, and thus settle the open problem raised at the end of that paper. Moreover, we investigate the bentness of the proposed binomials under the case $$\gcd (\frac{t}{2}, p^{\frac{n}{2}}+1)=1$$ gcd ( t 2 , p n 2 + 1 ) = 1 for some even integers $$t$$ t and $$n$$ n . Computer experiments show that the new class contains bent functions that are affinely inequivalent to known monomial and binomial ones. [ABSTRACT FROM AUTHOR]

In this paper, two-to-one mappings and involutions without any fixed point on finite fields of even characteristic are investigated. First, we characterize a closed relationship between them by implicit functions and develop an AGW-like criterion for 2-to-1 mappings. Using this criterion, some new constructions of 2-to-1 mappings are proposed and eight classes of 2-to-1 mappings of the form (x 2 k + x + δ) s + c x are obtained. Finally, a number of classes of involutions without any fixed point are derived from the known 2-to-1 mappings by the relation between them. [ABSTRACT FROM AUTHOR]

Are there other Boolean functions having two distinct Walsh coefficients except affine Boolean functions and maximal nonlinear (i.e. bent) Boolean functions? This paper proves that all Boolean functions with exactly two distinct Walsh coefficients are just the two known classes of affine and bent Boolean functions and the Boolean functions obtained by modifying the value of affine or bent Boolean functions at x = 0. [ABSTRACT FROM AUTHOR]

In the past few years, linear codes with few weights constructed from defining-sets have been extensively studied. In this paper, we further investigate a class of two-weight or three-weight linear codes from defining sets and determine their weight and complete weight enumerators by application of the theory of quadratic forms and some special Weil sums over finite fields. This paper generalizes some results in Jian et al. (2019), Yang (2020) and Zhu and Xu (2017). The parameters and weight distributions of the punctured codes of the discussed linear codes are also determined, and some optimal codes with respect to the Griesmer bound and some strongly regular graphs from two-weight projective codes are obtained. [ABSTRACT FROM AUTHOR]

Permutation polynomials over finite fields constitute an active research area and have important applications in many areas of science and engineering. In this paper, several classes of permutation polynomials with trace functions are presented over F p n (p = 2 , 3) by investigating the number of solutions to special equations. [ABSTRACT FROM AUTHOR]

Let m and k be two positive integers such that v 2 ( m ) = v 2 ( k ) , where v 2 ( ⋅ ) denotes the 2-valuation function, and let α be a primitive element of F p 2 m . Let C be a cyclic code over F p with a parity-check polynomial h 1 ( x ) h 2 ( x ) h 3 ( x ) ∈ F p 2 m [ x ] , where h 1 ( x ) , h 2 ( x ) and h 3 ( x ) are the minimal polynomials of α − p k + 1 2 , − α − p k + 1 2 and α − p m + 1 2 over F p respectively. In this paper, we determine the weight distribution and the complete weight distribution of the code C . [ABSTRACT FROM AUTHOR]

In this paper, we obtain several new classes of $$p$$ -ary bent functions, where $$p$$ is a prime. The bentness of all these functions is characterized by some exponential sums, which have close relations with Kloosterman sums. Moreover, we obtain some concise criterions on the bentness of $$p$$ -ary functions in some special cases. In addition, our work generalizes some main results obtained by Li et al. (IEEE Trans Inf Theory 59(3):1818-1831, ). [ABSTRACT FROM AUTHOR]