Classification of isomorphism classes of lattices from Construction A and B

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.