Applicable Algebra in Engineering Communication and Computing最新文献

We consider Noetherian operators in the context of symbolic computation. Upon utilizing the theory of holonomic ({mathcal D})-modules, we present a new method for computing Noetherian operators associated to a zero-dimensional ideal. An effective algorithm that consists mainly of linear algebra techniques is proposed for computing them. Moreover, as applications, new computation methods of polynomial ideals are discussed by utilizing the Noetherian operators.

The weight, balancedness and nonlinearity are important properties of Boolean functions, but they can be difficult to determine in general. In this paper, we study how to compute them for two classes of functions where these problems are more tractable. In particular, we study functions of degree three and the so-called “splitting” functions. The latter are functions that can be written as the sum of two functions defined over disjoint sets of variables. We show how, for splitting functions, studying these properties reduces to the study of simpler functions. We provide then a procedure to compute the weight of a cubic Boolean function. We show computationally that, for a cubic Boolean function with limited number of terms, this procedure is on average significantly more efficient than some other methods.

In this paper, using Gallai’s Theorem and the notion of strong resolving graph, we determine the strong metric dimension in annihilating-ideal graph of commutative rings. For reduced rings, an explicit formula is given and for non-reduced rings, under some conditions, strong metric dimension is computed.

Let C be an extremal Type III or IV code and (D_{w}) be the support design of C for weight w. We introduce the numbers, (delta (C)) and s(C), as follows: (delta (C)) is the largest integer t such that, for all weights, (D_{w}) is a t-design; s(C) denotes the largest integer t such that w exists and (D_{w}) is a t-design. Herein, we consider the possible values of (delta (C)) and s(C).

Maximum distance separable (MDS) codes have the highest possible error-correcting capability among codes with the same length and size. Let (gamma ) be nonzero in (mathbb {F}_{2^m}.) We consider all cyclic and ((1+ugamma ))-constacyclic codes of length (2^s) over (mathbb {F}_{2^m}+umathbb {F}_{2^m}) with their Lee distance and investigate all the cases whether the corresponding Gray images are MDS by giving an analogue of the Singleton bound for codes over (mathbb {F}_{2^m}+umathbb {F}_{2^m}) with the Lee distance through Gray map.

We describe software for symbolic computations that we developed in order to find Hamiltonian operators for Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations, and verify their compatibility. The computation involves nonlocal (integro-differential) operators, for which specific canonical forms and algorithms have been used.

In this work we define and study binary codes (C_{q,k}) and (overline{C_{q,k}}) obtained from neighborhood designs of Paley-type bipartite graphs P(q, k) and their complements, respectively for q an odd prime. We prove that for some values of q and k the codes ({C}_{q,k}) are self-dual and the codes (overline{C_{q,k}}) are self-orthogonal. Most of these codes tend to be with optimal or near optimal parameters. Next, we extend the codes (C_{q,k}) to get doubly even self dual codes and find that most of these codes are extremal.

Complementary decompositions of monomial ideals—also known as Stanley decompositions—play an important role in many places in commutative algebra. In this article, we discuss and compare several algorithms for their computation. This includes a classical recursive one, an algorithm already proposed by Janet and a construction proposed by Hironaka in his work on idealistic exponents. We relate Janet’s algorithm to the Janet tree of the Janet basis and extend this idea to Janet-like bases to obtain an optimised algorithm. We show that Hironaka’s construction terminates, if and only if the monomial ideal is quasi-stable. Furthermore, we show that in this case the algorithm of Janet determines the same decomposition more efficiently. Finally, we briefly discuss how these results can be used for the computation of primary and irreducible decompositions.

Linear codes with few weights have applications in cryptography, association schemes, designs, strongly regular graphs, finite group theory, finite geometries, and secret sharing schemes, among other disciplines. Two-weight linear codes are particularly interesting because they are closely related to objects in different areas of mathematics such as strongly regular graphs, 3-rank permutation groups, ovals, and arcs. There exist techniques to construct linear codes with few weights, for example, the systematic exposition by Calderbank and Kantor (Bull Lond Math Soc 18(2):97–122, 1986). Ding et al., (World Sci, pp 119–124, 2008) and (IEEE Trans Inf Theory 61(11):5835–5842, 2015) constructed few-weight codes using the trace function and Tonchev et al. (Algorithms, 12(8), 2019) generalized Ding’s construction. In this paper, we present an elementary way to get two- and three-weight codes from simplex codes and antipodal linear codes. An interesting application is the construction of uniformly packed linear codes from two-weight codes and quaternary quasi-perfect linear codes from three-weight codes.