  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.
