Applicable Algebra in Engineering Communication and Computing最新文献

In order to get a better code rate, this study focuses on the construction of double skew cyclic codes over the ring (textrm{R}= mathbb {F}_q+vmathbb {F}_q) with (v^2=v), where q is a prime power. We investigate the generator polynomials, minimal spanning sets, generator matrices, and the dual codes over the ring (textrm{R}). As an implementation, the obtained results are illustrated with some suitable examples. Here, we introduce a construction for new generator matrices and thus achieve codes with improved parameters compared to those available in the existing literature. Finally, we tabulate our obtained codes over the ring (textrm{R}).

The main results of this paper are in two directions. First, the family of finite local rings of length 4 whose annihilator of their maximal ideals have length 2 is determined. Second, the structure of constacyclic, reversible and DNA codes over those rings are described, the length of the code is relatively prime to the characteristic of the residue field of the ring.

Let ({mathfrak {R}}= {mathbb {Z}}_4[u,v]/langle u^2-2,uv-2,v^2,2u,2vrangle) be a ring, where ({mathbb {Z}}_{4}) is a ring of integers modulo 4. This ring ({mathfrak {R}}) is a local non-chain ring of characteristic 4. The main objective of this article is to construct reversible cyclic codes of odd length n over the ring ({mathfrak {R}}.) Employing these reversible cyclic codes, we obtain reversible cyclic DNA codes of length n, based on the deletion distance over the ring ({mathfrak {R}}.) We also construct a bijection (Gamma) between the elements of the ring ({mathfrak {R}}) and (S_{D_{16}}.) As an application of (Gamma ,) the reversibility problem which occurs in DNA k-bases has been solved. Moreover, we introduce a Gray map (Psi _{hom }:{mathfrak {R}}^{n}rightarrow {mathbb {F}}_{2}^{8n}) with respect to homogeneous weight (w_{hom }) over the ring ({mathfrak {R}}). Further, we discuss the GC-content of DNA cyclic codes and their deletion distance. Moreover, we provide some examples of reversible DNA cyclic codes.

Lattice theory has shown to be useful in information theory, and rotated lattices with high modulation diversity have been extensively studied as an alternative approach for transmission over a Rayleigh fading channel, where the performance depends on the minimum product distance to achieve coding gains. The maximum diversity of a rotated lattice is guaranteed when we use totally real number fields and the minimum product distance is optimized by considering fields with minimum discriminants. With the construction of full-diversity algebraic lattices as our goal, in this work we present and study constructions of full-diversity algebraic lattices in odd prime dimensional Euclidean spaces from families of modules in cyclic number fields. These families include all the ramified prime ideals in each of these number fields. As immediate applications of our results, we present algebraic constructions from the densest lattices in dimensions 3 and 5.

In this article, we deal with additive codes over the Frobenius ring ({mathcal {R}}_{2}{mathcal {R}}_{3}:=frac{{mathbb {Z}}_{2}[u]}{langle u^2 rangle }times frac{{mathbb {Z}}_{2}[u]}{langle u^3 rangle }). First, we study constacyclic codes over ({mathcal {R}}_2) and ({mathcal {R}}_3) and find their generator polynomials. With the help of these generator polynomials, we determine the structure of constacyclic codes over ({mathcal {R}}_2{mathcal {R}}_3). We use Gray maps to show that constacyclic codes over ({mathcal {R}}_{2}{mathcal {R}}_{3}) are essentially binary generalized quasi-cyclic codes. Moreover, we obtain a number of binary codes with good parameters from these ({mathcal {R}}_{2}{mathcal {R}}_{3})-constacyclic codes. Besides, several weight enumerators are computed, and the corresponding MacWilliams identities are established.

We describe the notion of stability of coherent systems as a framework to deal with redundancy. We define stable coherent systems and show how this notion can help the design of reliable systems. We demonstrate that the reliability of stable systems can be efficiently computed using the algebraic versions of improved inclusion-exclusion formulas and sum of disjoint products.

We study additive codes with 1-rank hulls and examine their properties for various dualities of the finite field of order 4. We give several constructions of additive and linear codes with 1-rank hulls. We also relate these codes to additive complementary dual codes (ACD). We give an interesting non-existence result for additive codes with a 1-rank hull for the duality (M_2) in terms of the parity of the number of generators. We conclude by giving substantive computations finding codes with one-rank hulls for small lengths using our results.

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).