Multidimensional Latin Bitrade

TL;DR

This paper characterizes small latin bitrades within hypercubes, explores their relation to t-fold MDS codes, and investigates embedding properties and component sizes, advancing combinatorial design theory.

Contribution

It provides a complete classification of small latin bitrades and analyzes their embedding into t-fold MDS codes, introducing new structural insights.

Findings

All admissible small cardinalities of latin bitrades are identified.

Symmetric difference of two 1-fold MDS codes always forms a latin bitrade.

Conditions for embedding latin bitrades into t-fold MDS codes are established.

Abstract

A subset $S$ of $k$-ary $n$-dimensional hypercube is called latin bitrade if $|S\cap F|\in\{0,2\} $ for each 1-face $F$. We find all admissible small (less than $2^{n+1}$) cardinalities of latin bitrades. A subset $M$ of $k$-ary $n$-dimensional hypercube is called $t$-fold MDS code if $|M\cap F|=t $ for each 1-face $F$. Symmetric difference of two 1-fold MDS codes is always a latin bitrade. Symmetric difference of two $t$-fold MDS codes may also be a latin bitrade. In this case we say that this latin bitrade embedded into $t$-fold MDS code. The intersection of $t$-fold MDS code and a latin bitrade embedded into it is called a component of the code. We study the questions of embedding of latin bitrades into $t$-fold MDS and admissible cardinalities of the component of $t$-fold MDS. Keywords: MDS code, latin bitrade, component.