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.

翻译：众所周知，格的离散模拟是线性码，它是汉明空间 $\mathbb{F}^n$ 的向量子空间。其中 $\mathbb{F}$ 是有限域，$n \in \mathbb{Z}_{>0}$。我们尝试构造一类格，使其离散模拟为变长非线性码。设 $\mathcal{G}$ 和 $\mathcal{H}$ 为两个有限群，$\mathcal{S}$ 为 $\mathcal{G}$ 的固定生成元集合。同态码定义为从 $\mathcal{G}$ 到 $\mathcal{H}$ 的所有同态的集合，记为 $\mathcal{C} = Hom(\mathcal{G}, \mathcal{H})$。对每个 $\mathcal{G}$ 与 $\mathcal{H}$ 之间的同态 $\varphi$，关联一个码字 $c_\varphi$，它是 $\varphi$ 在生成元集合 $\mathcal{S}$ 上的取值向量，即 $c_\varphi = (\varphi(s_1), \varphi(s_2), \dots, \varphi(s_k))$，其中 $\varphi(s_i)$ 是 $s_i \in \mathcal{S}$ 的像，$1 \leq i \leq k$。我们提出一种设计方案，从秩大于1的有限阿贝尔 $p$-群（$p$ 为素数）构造一类称为自同构轨道码的变长二元非线性码。对每个有限阿贝尔 $p$-群，自同构轨道码的码字称为自同构轨道码字，其长度可变。注意，同态码由群间的同态决定，而自同构轨道码则由数的划分、群作用的轨道、群同态与自同构共同描述。我们利用 $Hom(\mathcal{G}, \mathcal{H})$ 中的元素为自同构轨道码的码字比特串建立覆盖关系，并构造变长非线性码的格。最后，我们讨论与群和代数的表示理论相关的未来研究方向。