Refinements of Levenshtein bounds in $q$-ary Hamming spaces

TL;DR

This paper improves Levenshtein bounds for q-ary Hamming spaces by incorporating discrete distance considerations, leading to new bounds and analogs of known bounds, with efficient calculations and potential code parameters.

Contribution

It introduces refined bounds for q-ary Hamming spaces that account for discrete distances, generalizes MacEliece bounds, and offers efficient computation methods.

Findings

New bounds for q-ary Hamming spaces derived

Generalizations of MacEliece bounds presented

Efficient calculation methods demonstrated

Abstract

We develop refinements of the Levenshtein bound in $q$-ary Hamming spaces by taking into account the discrete nature of the distances versus the continuous behavior of certain parameters used by Levenshtein. The first relevant cases are investigated in detail and new bounds are presented. In particular, we derive generalizations and $q$-ary analogs of a MacEliece bound. We provide evidence that our approach is as good as the complete linear programming and discuss how faster are our calculations. Finally, we present a table with parameters of codes which, if exist, would attain our bounds.