Classifying Subspaces of Hamming Spaces

Abstract

A linear code in Fnq with dimension k and minimum distance at least d is called an ln, k, drq code. We here consider the problem of classifying all ln, k, drq codes given n, k, d, and q. In other words, given the Hamming space Fnq and a dimension k, we classify all k-dimensional subspaces of the Hamming space with minimum distance at least d. Our classification is an iterative procedure where equivalent codes are identified by mapping the code equivalence problem into the graph isomorphism problem, which is solved using the program nauty. For d e 3, the classification is explicitly carried out for binary codes of length n ≤ 14, ternary codes of length n ≤ 11, and quaternary codes of length n ≤ 10.