Classification of isomorphism classes of lattices from Construction A and B

TL;DR

This paper classifies when lattices from Construction A and B, derived from self-orthogonal codes over finite fields, are isomorphic, linking lattice isomorphisms to code isomorphisms.

Contribution

It provides a complete classification of isomorphism classes of certain lattices from Construction A and B based on code isomorphisms, extending previous classifications.

Findings

Lattices from Construction A and B are isomorphic iff the underlying codes are isomorphic.

The classification relates lattice isomorphisms to code isomorphisms for self-orthogonal codes.

Generalizes the notion of a lattice frame and introduces new codes analogous to Kleinian codes.

Abstract

In this paper, we completely classify the isomorphism classes of certain lattices $L_A(C)$ and $L_B(C)$ from a self-orthogonal code $C$ over the finite field $\mathbb{F}_p$, where $p$ is an odd prime. These lattices are obtained by \emph{Construction A} and \emph{B} for a code $C$ over $\mathbb{F}_p$ introduced by Lam and Shimakura, which arose from a study of orbifolds of lattice vertex operator algebras. For self-orthogonal codes $C$ and $D$ of the same length over $\mathbb{F}_p$, we show that $L_X(C) \cong L_X(C)$ as lattices if and only if $C \cong D$ as codes, where $X=A$ or $B$. This can be expected to be lattice analogues of classifications of the isomorphism classes of lattice vertex operator algebras and its orbifolds. To prove the result, we generalize the notion of a frame of a lattice and define some codes which are analogues of codes constructed from Kleinian codes studied by H{\"o}hn.