Codes and Orbit Covers of Finite Abelian Groups

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.