Graphs Associated with Codes of Covering Radius 1 and Minimum Distance 2

Abstract

The search for codes of covering radius $1$ led Östergård, Quistorff and Wassermann to the OQW method of associating a unique graph to each code. We present results on the structure and existence of OQW-associated graphs. These are used to find an upper bound on the size of a ball of radius $1$ around a code of length $3$ and minimum distance $2$. OQW-associated graphs and non-extendable partial Latin squares are used to catalogue codes of length $3$ over $4$ symbols with covering radius $1$ and minimum distance $2$.