Applicable Algebra in Engineering Communication and Computing最新文献

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.

An almost perfect non-linear (APN) function over (mathbb {F}_{2^n}) is called exceptional APN if it remains APN over infinitely many extensions of (mathbb {F}_{2^n}). Exceptional APN functions have attracted attention of many researchers in the last decades. While the classification of exceptional APN monomials has been done by Hernando and McGuire, it has been conjectured by Aubry, McGuire and Rodier that up to equivalence, the only exceptional APN functions are the Gold and the Kasami–Welch monomial functions. Since then, many partial results have been on classifying non-exceptional APN polynomials. In this paper, for the classification of the exceptional property of APN functions, we introduce a new method that uses techniques from curves over finite fields. Then, we apply the method with Eisenstein’s irreducibility criterion and Kummer’s theorem to obtain new non-exceptional APN functions.

Low-Rank Parity-Check (LRPC) codes are a class of rank metric codes that have many applications specifically in network coding and cryptography. Recently, LRPC codes have been extended to Galois rings which are a specific case of finite rings. In this paper, we first define LRPC codes over finite commutative local rings, which are bricks of finite rings, with an efficient decoder. We improve the theoretical bound of the failure probability of the decoder. Then, we extend the work to arbitrary finite commutative rings. Certain conditions are generally used to ensure the success of the decoder. Over finite fields, one of these conditions is to choose a prime number as the extension degree of the Galois field. We have shown that one can construct LRPC codes without this condition on the degree of Galois extension.

In complex real-world networks, the relation among vertices (people) changes over time. Even with millions of vertices, adding new vertices or deleting a few previous ones can drastically change the network’s dynamics. The Iterated Local Transitivity model is a deterministic model based on the principle of transitivity and local interaction among people. The same has been extended to signed social networks. Let (Sigma) be a signed graph with underlying graph (G = (V, E)) and a function (sigma :Erightarrow {+,-}) assigning signs to the edges. We determine the relation between the characteristic polynomials of signed graph (Sigma) and the signed graph obtained from (Sigma) by adding (deleting) vertices or by adding (deleting) edges. Consequently, we present a recurrence relation for a characteristic polynomial of the Iterated Local Transitivity model for signed graphs.