One-Sided $k$-Orthogonal Matrices Over Finite Semi-Local Rings And Their Codes

TL;DR

This paper studies one-sided $k$-orthogonal matrices over finite semi-local rings, exploring their algebraic structures, constructing new matrix semigroups, and connecting these matrices to the theory of self-dual codes.

Contribution

It introduces the properties and semigroup structures of one-sided $k$-orthogonal matrices over finite semi-local rings and links these matrices to self-dual linear codes.

Findings

Matrices form semigroups when $k$ is idempotent.

Semigroups are isomorphic to products over fields for semi-local rings.

Connections established between matrices and self-dual codes.

Abstract

Let $R$ be a finite commutative ring with unity $1_R$ and $k \in R$. Properties of one-sided $k$-orthogonal $n \times n$ matrices over $R$ are presented. When $k$ is idempotent, these matrices form a semigroup structure. Consequently new families of matrix semigroups over certain finite semi-local rings are constructed. When $k=1_R$, the classical orthogonal group of degree $n$ is obtained. It is proved that, if $R$ is a semi-local ring, then these semigroups are isomorphic to a finite product of $k$-orthogonal semigroups over fields. Finally, the antiorthogonal and self-orthogonal matrices that give rise to leading-systematic self-dual or weakly self-dual linear codes are discussed.