Applicable Algebra in Engineering Communication and Computing最新文献

As an important class of cyclic codes, BCH codes are widely employed in satellite communications, DVDs, CD, DAT etc. In this paper, we determine the dimension of BCH codes of length (frac{q^{2m}-1}{q+1}) over the finite fields ({mathbb {F}}_q). We settle a conjecture about the largest q-cyclotomic coset leader modulo n which was proposed by Wu et al. We also get the second largest q-cyclotomic coset leader modulo n if m is odd. Moreover, we investigate the parameters of ({mathcal {C}}_{(n,q,delta _i)}) (({mathcal {C}}_{(n,q,delta _i)}^perp)) for (i=1,2) and ({mathcal {C}}_{(n,q,delta )}) (({mathcal {C}}_{(n,q,delta )}^perp)) for (2le delta le q-1). What’s more, we obtain many (almost) optimal codes.

Linear codes are widely studied due to their important applications in authentication codes, association schemes and strongly regular graphs, etc. In this paper, a class of at most three-weight linear codes is constructed by selecting a new defining set, then the parameters and weight distributions of codes are determined by exponential sums. Results show that almost all the linear codes we constructed are minimal and we also describe the access structures of the secret sharing schemes based on their dual. Especially, the new binary code is a two-weight projective code and based on which a strongly regular graph with new parameters is designed.

As a class of linear codes, cyclic codes are widely used in communication systems, consumer electronics and data storage systems due to their favorable properties. In this paper, we construct two classes of optimal p-ary cyclic codes with parameters ([p^m-1, p^m-frac{3m}{2}-2, 4]) by analyzing the solutions of certain polynomials over finite fields. Furthermore, we propose an efficient method to determine the optimality of the 7-ary cyclic code (mathcal {C}_{(0,1,e)}) and present three classes of optimal codes with parameters ([7^m-1,7^m-2m-2,4]). Additionally, we provide the weight distribution of one class of their duals.

At present, GLV/GLS scalar multiplication mainly focuses on elliptic curves in Weierstrass form, attempting to find and construct more and more efficiently computable endomorphism. In this paper, we investigate the application of the GLV/GLS scalar multiplication technique to Jacobi Quartic curves. Firstly, we present a concrete construction of efficiently computable endomorphisms for this type of curves over prime fields by exploiting birational equivalence between curves, and obtain a 2-dimensional GLV method. Secondly, we consider the quadratic twists of elliptic curves. By using birational equivalence and Frobenius mapping between curves, we present methods to construct efficiently computable endomorphisms for this type of curves over the quadratic extension field, and obtain a 2-dimensional GLS method. Finally, we obtain a 4-dimensional GLV method on elliptic curves with j-invariant 0 or 1728 by using higher degree twists. The experimental results show that the speedups of the 2-dimensional GLV method and 4-dimensional GLV method compared to 5-NAF method exceed (37.2%) and (109.4%) for Jacobi Quartic curves, respectively. At the same time, when utilizing one of the proposed methods, the scalar multiplication on Jacobi Quartic curves is consistently more efficient than on elliptic curves in Weierstrass form.

We present a new method of constructing three dimensional periodic arrays by composing a two dimensional periodic array with a sequence of shifts consisting of a cyclic group of points on an elliptic curve over a prime field ({mathbb {F}}_p). For every base array B with period (c, c) and every cyclic group G of order C there are (phi (C)) families, each of size (p^2), of 3D arrays with period (c, c, C). We illustrate our method using a Legendre array as base array. The resulting 3D arrays have period (p, p, C), peak auto-correlation value (C(p^2-1)), and non-peak auto-correlation and cross-correlation values of the form (kp^2-C) where C is the order of the group and, in the general case, (kle 3). We present experimental results that show that (kle 2) for a certain type of cyclic group of points in ({mathbb {F}}_p) when (p<1000). Finally, we show that the linear complexity of our constructions compare favorably with other known constructions.

We introduce a new type of characteristic sets of difference polynomials using a generalization of the concept of effective order to the case of partial difference polynomials and a partition of the basic set of translations (sigma). Using properties of these characteristic sets, we prove the existence and outline a method of computation of a multivariate dimension polynomial of a finitely generated difference field extension that describes the transcendence degrees of intermediate fields obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of (sigma) are bounded by two sequences of natural numbers. We show that such dimension polynomials carry essentially more invariants (that is, characteristics of the extension that do not depend on the set of its difference generators) than previously known difference dimension polynomials. In particular, a dimension polynomial of the new type associated with a system of algebraic difference equations gives more information about the system than the classical univariate difference dimension polynomial.

Modular strongly regular graphs have been introduced by Greaves et al. (Linear Algebra Appl 639:50–80, 2022). We show that a related class of isodual codes is asymptotically good. Equiangular tight frames over finite fields also introduced by the same authors in 2022 are shown here to connect with self-dual codes. We give several examples of equiangular tight frames over finite fields arising from self-dual codes.

Functions with low differential uniformity have wide applications in cryptography. In this paper, by using the quadratic character of ({mathbb {F}}_{p^n}^*), we further investigate the ((-1))-differential uniformity of these functions in odd characteristic: (1) (f_1(x)=x^d), where (d=-frac{p^n-1}{2}+p^k+1), n and k are two positive integers satisfying (frac{n}{gcd (n,k)}) is odd; (2) (f_2(x)=(x^{p^m}-x)^{frac{p^n-1}{2}+1}+x+x^{p^m}), where (n=3m); (3) (f_3(x)=(x^{3^m}-x)^{frac{3^n-1}{2}+1}+(x^{3^m}-x)^{frac{3^n-1}{2}+3^m}+x), where (n=3m). The results show that the upper bounds on the ((-1))-differential uniformity of the power function (f_1(x)) are derived. Furthermore, we determine the ((-1))-differential uniformity of two classes of permutation polynomials (f_2(x)) and (f_3(x)) over ({mathbb {F}}_{p^n}) and ({mathbb {F}}_{3^n}), respectively. Specifically, a class of permutation polynomial (f_3(x)) that is of P(_{-1})N or AP(_{-1})N function over ({mathbb {F}}_{3^n}) is obtained.

In this paper, we propose a novel method for constructing maximum distance separable (MDS) codes based on the extended invertibility and orthogonality of quasigroups. We provide various methods of constructing an orthogonal system of k-ary operations over (Q^2) using a special type of k-ary operations over Q, where Q is any arbitrary finite set. Then we use concepts of strong orthogonality of k-ary operations to establish a connection between orthogonality and linear recursive MDS codes. We illustrate these new classes of MDS codes using the proposed techniques and enumerate such codes using SageMath.

The notion of (k-)th order nonlinear complexity has been studied from various aspects. Geil, Özbudak and Ruano (Semigroup Forum 98:543–555, 2019) gave a construction of a sequence of length ((q-1)(q^{2}-1)) with high nonlinear complexity by using the Weierstrass semigroup of two distinct rational points on a Hermitian function field over (F_{q^{2}}), and they improved the bounds on the (k-)th order nonlinear complexity (N^{k}(s)) and (L^{k}(s)) obtained by Niederreiter and Xing (IEEE Trans Inf Theory 60(10):6696–6701, 2014), where (F_{q^2}) is the finite field with (q^2) elements, and q is a prime power. In this work, we exhibit the lower bounds on (N^{k}(s)) and (L^{k}(s)) on a Hermitian function field using Hermitian triangles over (F_{q^2}.) We study the effect of a Hermitian triangle by its type. The possible cases on the k-th order nonlinear complexity are explained, for each type, and we improve the lower bounds obtained by Geil et al. We construct two different sequences with the help of the Weierstrass semigroup of l distinct collinear rational points, and we compare our results of the lower bounds on (N^{k}(s)) and (L^{k}(s).) Also, we study the lower bounds on (N^{k}(s)) and (L^{k}(s)) using the Weierstrass semigroup of two distinct rational points on a Suzuki function field over (F_{q},) where (q={2q_{0}}^{2}, q_{0}=2^{t}, tge 1.)