Applicable Algebra in Engineering Communication and Computing最新文献

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 (mathbb {F}_{2^n}) by using the multivariate method and resultant elimination, and show that these 0-APN power functions are CCZ-inequivalent to the known ones.

Abstract

In this paper, we consider the finite groups $$begin{aligned} H_{(t,l,m)}=langle a,b,c | a^t=b^l=c^m=1, [a,b]=c, [a,c]=[b,c]=1rangle . end{aligned}$$ We obtain the generalized order-k Pell sequences and study their periods. We prove the period of the order-k Pell sequence divided the period of the generalized order-k Pell sequence in the Heisenberg group. Then, the generalized order-k Pell sequence in Heisenberg group are used to define new cyclic groups. As an application, these groups are used in encryption algorithms.

This paper focuses on providing the characteristics of generalized Boolean functions from a new perspective. We first generalize the classical Fourier transform and correlation spectrum into what we will call the (rho)-Walsh–Hadamard transform ((rho)-WHT) and the (rho)-correlation spectrum, respectively. Then a direct relationship between the (rho)-correlation spectrum and the (rho)-WHT is presented. We investigate the characteristics and properties of generalized Boolean functions based on the (rho)-WHT and the (rho)-correlation spectrum, as well as the sufficient (or also necessary) conditions and subspace decomposition of (rho)-bent functions. We also derive the (rho)-autocorrelation for a class of generalized Boolean functions on ((n+2))-variables. Secondly, we present a characterization of a class of generalized Boolean functions with (rho)-WHT in terms of the classical Boolean functions. Finally, we demonstrate that (rho)-bent functions can be obtained from a class of composite construction if and only if (rho =1). Some examples of non-affine (rho)-bent functions are also provided.

Based on the inverse Gray mapping and sign alternation transform, a new family of quaternary sequences with optimal odd-periodic autocorrelation magnitude has been constructed by using the Legendre sequence pair, twin-prime sequence pair and GMW sequence pair. In this paper, we use the correlation properties of the Legendre sequence pair, twin-prime sequence pair and GMW sequence pair to determine the lower bound of 4-adic complexity of these quaternary sequences, as well as show that these quaternary sequences have large 4-adic complexity.

Let (n=2(p^m-1)/(p-1)), where p is an odd prime and (m>1) is a positive integer. In this paper, we research optimal p-ary constacyclic codes with two zeros. Two classes of optimal p-ary ([n,n-2m,4]) constacyclic codes are presented by searching the solutions of certain congruence equations over (mathbb {F}_{p^m}). Four explicit constructions of optimal constacyclic codes with such parameters are provided. The dual codes of a subclass of these constacyclic codes are also investigated.

In this paper, we determine the differential spectra of two known classes of involutions with low differential uniformity over finite fields with even characteristic completely. The key point of our method is that we propose several new definitions called special and ordinary points. In addition, it is interesting that one of our differential spectra is relative to the well-known Kloosterman sum.

This paper extends the concept of weighted point clouds and weighted simplicial complexes by introducing product point clouds and product simplicial complexes within a commutative ring with unity. Within an integral domain, the introduction of a weighted product chain group, along with the induced product weighted homomorphism and weighted product boundary maps, leads to significant outcomes and findings. To explore the algebraic characteristics of a weighted product structure, we introduce the concept of weighted product homology. This homology considers the relationship of weights assigned to elements within the structure and their impact on the structure’s underlying algebraic properties.

The weight hierarchy of a linear code have been an important research topic in coding theory since Wei’s original work in 1991. In this paper, choosing (D=Big {(x,y)in Big ({mathbb {F}}_{p^{s_1}}times {mathbb {F}}_{p^{s_2}}Big )Big backslash {(0,0)}: f(x)+text {Tr}_1^{s_2}(alpha y)=0Big }) as a defining set, where (alpha in {mathbb {F}}_{p^{s_2}}^*), f(x) is a quadratic form over ({mathbb {F}}_{p^{s_1}}) with values in ({mathbb {F}}_p) and f(x) can be non-degenerate or not, we construct a family of three-weight p-ary linear codes and determine their weight distributions and weight hierarchies completely. Most of the codes can be used in secret sharing schemes.

In this paper, a necessary condition which is sufficient as well for a pair of constacyclic 2-D codes over a finite commutative ring R to be an LCP of codes has been obtained. Also, a characterization of non-trivial LCP of constacyclic 2-D codes over R has been given and total number of such codes has been determined. The above results on constacyclic 2-D codes have been extended to constacyclic 3-D codes over R. The obtained results readily extend to constacyclic n-D codes, (n ge 3), over finite commutative rings. Furthermore, some results on existence of non-trivial LCP of constacyclic 2-D codes over a finite chain ring have been obtained in terms of its residue field.