Journal of Algebra Combinatorics Discrete Structures and Applications

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

Norifumi Ojiro — 2025-04-28

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

Fidel Ochieng Oduol — 2025-04-28

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.

New Results and Bounds on Codes over GF(19)

Saurav Pandey — 2025-08-31

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).

On the three graph invariants related to matching of finite simple graphs

Kazunori Matsuda — 2025-08-31

Let G be a finite simple graph on the vertex set V(G) and let ind-match(G), min-match(G) and match(G) denote the induced matching number, the minimum matching number and the matching number of G, respectively. It is known that the inequalities ind-match(G) <= min-match(G) <= match(G) <= 2min-match(G) and match(G) <= |V(G)|/2 hold in general.

In the present paper, we determine the possible tuples (p, q, r, n) with ind-match(G) = p, min-match(G) = q, match(G) = r and |V(G)| = n arising from connected simple graphs. As an application of this result, we also determine the possible tuples (p`, q, r, n) with reg(G) = p`, min-match(G) = q, match(G) = r and |V(G)| = n arising from connected simple graphs, where I(G) is the edge ideal of G and reg(G) = reg(K[V(G)]/I(G)) is the Castelnuovo--Mumford regualrity of the quotient ring K[V(G)]/I(G).

Local System of Simple Locally Finite Associative Algebras

Hasan M. S. Shlaka — 2025-02-06

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 briefly

On weakly S-2-absorbing filters of lattices

Shahabaddin Ebrahimi Atani — 2025-08-31

Let £ be a bounded distributive lattice and S a join closed subset of £. Following the concept of weakly S-2-absorbing submodules, we define weakly S-2-absorbing filters of £. We will make an intensive investigate the basic properties and possible structures of these filters.

On additive cyclic codes over F_4+uF_4

Gyanendra K. Verma — 2025-04-28

This article studies additive cyclic codes over R = F4 + uF4, where u^2 = 0. We obtain generator polynomials for these codes and provide necessary
and 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.

A note on CII groups and CCII groups

Teerapong Suksumran — 2025-04-28

A group G is CII or, equivalently, 2-Engel if [g, h]= [g^-1;h^-1] for all elements g and h in G, and is CCII if the central quotient G/Z(G) is CII. In this paper, we give sufficient conditions and necessary conditions for a group to be CCII. In particular, we show that every CCII group is nilpotent of class at most 4 and list all CII groups and all CCII groups of order n with n < 64 up to isomorphism.

Local and 2-local 1/2-derivation on naturally graded non-Lie p-filiform algebras

Baxtiyor Yusupov — 2025-08-31

This paper is devoted to study local and 2-local 1/2-derivation on p-filiform Leibniz algebras. We prove that p-filiform Leibniz algebras as a rule admit local(2-local) 1/2-derivations which are not 1/2-derivations.