Orbit codes of finite Abelian groups and lattices

This paper constructs a class of lattices whose discrete analogues are variable-length nonlinear codes. The well-known discrete analogue of lattices and linear codes inspires our approach. We next design a variable length binary non -linear code called automorphism orbit code from a finite abelian p-group of rank greater than 1, where p is a prime. For each finite abelian p-group, codewords of the automorphism orbit code are variablelength codewords called automorphism orbit codewords. The homomorphisms between groups determine homomorphism codes, whereas automorphism orbit codes are specified by partitions of a number, orbits of group action, homomorphisms and automorphisms of groups. For some groups G and 7-l, we shall use elements of Hom(G, 7-l) to create a cover relation for bit strings of codewords of an automorphism orbit code and formulate a lattice of variable length non -linear codes. (c) 2024 Elsevier B.V. All rights reserved.