New Results and Bounds on Codes over GF(19)

Explicit construction of linear codes over finite fields is one of the most important and challenging problems in coding theory. Due to the centrality of this problem, databases of best-known linear codes (BKLCs) over small finite fields have been available. Recently, new databases for BKLCs over larger alphabets have been introduced. In this work, a new database of BKLCs over the field  GF(19)  is introduced, containing lower and upper bounds on the minimum distances for codes with lengths up to  150  and dimensions between  3  and  6. Computer searches were conducted on cyclic, constacyclic, quasi-cyclic, and quasi-twisted codes to establish lower bounds. These searches resulted in many new linear codes over  GF(19)

Adjacency spectrum and Wiener index of essential ideal graph of a finite commutative ring

Let R be a commutative ring with unity. The essential ideal graph ER of R, is a graph with a vertex set consisting of all nonzero proper ideals of R and two vertices I and K are adjacent if and only if I + K is an essential ideal. In this paper, we study the adjacency spectrum of the essential ideal graph of the finite commutative ring Zn, for n = {pm, pm1qm2}, where p, q are distinct primes, and m, m1 , m2 ε N.  We show that 0 is an eigenvalue of the adjacency matrix of EZn if and only if either n = p2 or n is not a product of distinct primes. We also determine all the eigenvalues of the adjacency matrix of EZn whenever n is a product of three or four distinct primes. Moreover, we calculate the topological indices, namely the Wiener index and hyper-Wiener index of the essential ideal graph of Zn, for different forms of n

Valuation overrings of polynomial rings and group of divisibility

In this work, we discuss the types of valuation overrings of $K[x_{1}, x_{2}, ..., x_{n}]$ based on the rank and rational rank of value groups. Also, we describe the group of divisibility of a finite intersection of valuation overrings of $K[x_{1}, x_{2}, ..., x_{n}].$ In particular, we focus on the case for $n > 3.

On additive cyclic codes over F_4+uF_4

This article studies additive cyclic codes over R = F4 + uF4, where u^2 = 0. We obtain generator polynomials for these codes and provide necessaryand sufficient conditions for additive codes to be self-orthogonal and self-dual codes over R with respect to the symplectic inner product. Additive self-orthogonal codes over F4 with respect to the symplectic inner product are used to construct quantum codes. We demonstrate that the Gray image of additive self-orthogonal codes over R results in additive self-orthogonal codes over F4. Additionally, we prove that binary self-orthogonal codes can be obtained from additive self-orthogonal codes over R with respect to the symplectic inner product

Totally projective QTAG-modules and generalizations

In this project, we prioritize our study on some types of generalized torsion abelian groups. The torsion abelian group is an important tool in the theory of modules. Analogous to this concept, we study the totally projective modules and discuss its relation with isotype as well as separable submodules. One of the main purposes of the present paper is to give a necessary and sufficient condition for an isotype submodule of a totally projective module to be itself a totally projective module

On generator polynomial matrices of quasi-cyclic codes with linear complementary duals

Using notion of generator polynomial matrices of quasi-cyclic codes, we show a necessary and sufficient condition for which these codes are to be linear complementary dual. This extends the well-known result by Yang and Massey on cyclic codes to quasi-cyclic codes. As an application we present various examples of optimal binary LCD quasi-cyclic codes

On a variant of k-plane trees

In this paper, we introduce a class of plane trees whose vertices receive labels from the set {1,2,...,k} such that the sum of labels of adjacent vertices does not exceed k+1 and all vertices of label 1 are always on the left of all other vertices. Using generating functions, we enumerate these trees by number of vertices and label of the root, root degree, label of the eldest or youngest child of the root and forests. &nbsp

θ\theta-Generalized monomial codes

In this paper we generalize cyclic codes to another more large linear codes, that is $\theta$-monomial codes. It is shown that for a $\theta$-monomial code, its Euclidean and $e$-Galois dual is also  $\theta$-monomial code. Furthermore we present the equivalence between $\theta$-monomial codes and generalized monomial codes. By considering the skew polynomial ring, we show that $\theta$-monomial codes can relate to submodules under a condition and to ideals under other condition,this allow us to give a characterization of $\theta$-monomial codes. More results on the $e$-Galois dual of $\theta$-monomial codes are given with additional properties on self duality and self orthogonality. The Generalized $\theta$-monomial codes are discussed with their algebraic structure. The paper is closed by the investigation of the algebraic structure of $\theta$-monomial codes over the ring $\mathbb{F}_q+v\mathbb{F}_q$ where $v^2=v$.  &nbsp

On submodule spectrum in multiplication le-modules

In this article, we have studied the Zariski topology related to a submodule element of a le-module. Obtained a base for the complement of the submodule spectrum and topological features, along with some characterizations of the radical of a submodule element, are established. Several algebraic conditions are obtained for an open subset concerning the Zariski topology to become compact, dense, Noetherian, etc

Local System of Simple Locally Finite Associative Algebras

Abstract. In this paper, we study local systems of locally finite associative algebras over fields of characteristic p\ge0. We describe the perfect local systems and study the relation between them and their corresponding locally finite associative algebras. 1-perfect and conical local systems are also be considered and described briefl