Applicable Algebra in Engineering Communication and Computing最新文献

In this paper, we give a construction of doubly even self-orthogonal codes from quasi-symmetric designs. Especially, we study orbit matrices of quasi-symmetric designs and give a construction of doubly even self-orthogonal codes from orbit matrices of quasi-symmetric designs of Blokhuis–Haemers type.

Let ({{mathbb {K}}}) be any field, let (Xsubset {mathbb P}^{k-1}) be a set of (n) distinct ({{mathbb {K}}})-rational points, and let (age 1) be an integer. In this paper we find lower bounds for the minimum distance (d(X)_a) of the evaluation code of order (a) associated to (X). The first results use (alpha (X)), the initial degree of the defining ideal of (X), and the bounds are true for any set (X). In another result we use (s(X)), the minimum socle degree, to find a lower bound for the case when (X) is in general linear position. In both situations we improve and generalize known results.

Cyclic codes over finite fields have been studied for decades due to their wide applicability in communication systems, consumer electronics, and data storage systems. Let p be an odd prime and let s and m be positive integers. In this paper, we first determine the Hamming distances of all cyclic codes of length 8 over (F_q). Building upon this, we explicitly obtain the Hamming distances of all repeated-root cyclic codes of length (8p^s) over (F_q). As an application, we give all maximum distance separable cyclic codes of length (8p^s).

Permutation polynomials with sparse forms over finite fields attract researchers’ great interest and have important applications in many areas of mathematics and engineering. In this paper, by investigating the exponents (s, i) and the coefficients (a,bin mathbb {F}_{q}^{*}), we present some new results of permutation polynomials of the form (f(x)= ax^{q^i(q^{2}-q+1)} + bx^{s} + textrm{Tr}_{q^n/q}(x)) over (mathbb {F}_{q^n}) ((n=2) or 3). The permutation property of the new results is given by studying the number of solutions of special equations over (mathbb {F}_{q^n}).

Let (k>1) be a fixed integer. In Gupta and Ray (Cryptography and Communications 7: 257–287, 2015), proved the non existence of (2^k times 2^k) orthogonal circulant MDS matrices and involutory circulant MDS matrices over finite fields of characteristic 2. The main aim of this paper is to prove the non-existence of orthogonal circulant MDS matrices of order (2^ktimes 2^k) and involutory circulant MDS matrices of order k over finite commutative rings of characteristic 2. Precisely, we prove that any circulant orthogonal matrix of order (2^k) over finite commutative rings of characteristic 2 with identity is not a MDS matrix. Moreover, some related results are also discussed. Finally, we provide some examples to prove that the assumed restrictions on our main results are not superfluous.

Consider a multivariate polynomial (f in K [x_1, ldots , x_n]) over a field K, which is given through a black box capable of evaluating f at points in (K^n), or possibly at points in (A^n) for any K-algebra A. The problem of sparse interpolation is to express f in its usual form with respect to the monomial basis. We analyze the complexity of various old and new algorithms for this task in terms of bounds D and T for the total degree of f and its number of terms. We mainly focus on the case when K is a finite field and explore possible speed-ups.

Several problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptosystems. A finite chain ring is a finite ring admitting exactly one maximal ideal and every ideal being generated by one element. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residue field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gröbner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Plücker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.

Let R be a commutative ring with identity and A(R) be the set of ideals with non-zero annihilator. The annihilator-ideal graph of R is defined as the graph (mathrm{A_I}(R)) with the vertex set (A(R)^*=A(R)setminus {0}) and two distinct vertices L, K are adjacent if and only if (textrm{Ann}_R(K) cup textrm{Ann}_R(L)) is a proper subset of (textrm{Ann}_R(KL)). In this paper, we determine the metric dimension of (mathrm{A_I}(R)). Also, the twin-free clique number for (mathrm{A_I}(R)) is computed and as an application the strong metric dimension in annihilator-ideal graphs is given.

Let C be a two and three-weight ternary code. Furthermore, we assume that (C_ell ) are t-designs for all (ell ) by the Assmus–Mattson theorem. We show that (t le 5). As a corollary, we provide a new characterization of the (extended) ternary Golay code.

In this paper we use the puncturing and shortening techniques on two already-known classes of optimal cyclic codes in order to obtain three new classes of optimal linear codes achieving the Griesmer bound. The weight distributions for these codes are settled. We also investigate their dual codes and show that they are either optimal or almost optimal with respect to the sphere-packing bound. Moreover, these duals contain classes of almost maximum distance separable codes which are shown to be proper for error detection. Further, some of the obtained optimal linear codes are suitable for constructing secret sharing schemes with nice access structures.