Codes and Orbit Covers of Finite Abelian Groups

TL;DR

This paper introduces a novel class of variable length, non-linear codes derived from finite abelian p-groups, utilizing group homomorphisms, automorphisms, and group actions to construct and analyze their structure.

Contribution

It constructs automorphism orbit codes from finite abelian p-groups and develops a lattice framework for these variable length non-linear codes, connecting group theory with coding theory.

Findings

Automorphism orbit codes are variable length and non-linear.

A lattice structure for these codes is formulated.

Connections to group representation theory are discussed.

Abstract

It is well known that the discrete analogue of a lattice is a linear code which is a vector subspace of Hamming space $\mathbb{F}^n$. The set $\mathbb{F}$ is a finite field and $n \in \mathbb{Z}_{>0}$. Our attempt is to construct a class of lattices such that its discrete analogues are variable length non-linear codes. Let $\mathcal{G}$ and $\mathcal{H}$ be two finite groups, and let $\mathcal{S}$ be a fixed set of generators for $\mathcal{G}$. The homomorphism code is defined as the set of all homomorphisms from $\mathcal{G}$ to $\mathcal{H}$, denoted by, $\mathcal{C} = Hom(\mathcal{G}, \mathcal{H})$. To each homomorphism $\varphi$ between $\mathcal{G}$ and $\mathcal{H}$, a codeword $c_\varphi$ is associated, it is a vector of values of $\varphi$ on the generators in $\mathcal{S}$, that is, $c_\varphi = (\varphi(s_1), \varphi(s_2), \dots, \varphi(s_k))$, where $\varphi(s_i)$ is the image of $s_i \in \mathcal{S}$, $1 \leq i \leq k$. We provide a design to construct a variable length binary non-linear code called as automorphism orbit code from a finite abelian $p$-group of rank more than 1, where $p$ is a prime number. For each finite abelian $p$-group, the codewords of the automorphism orbit code are variable length codewords called as automorphism orbit codewords. Note that homomorphism codes are determined by homomorphisms between groups, whereas automorphism orbit codes are specified by partitions of a number, orbits of a group action, homomorphisms and automorphisms of groups. We make use of elements of $Hom(\mathcal{G}, \mathcal{H})$ to present a cover relation for bit strings of codewords of an automorphism orbit code and formulate a lattice of variable length non-linear codes. Finally, we discuss some information related to the future research work on connections to representation theory of groups and algebras.