Non-Binary Diameter Perfect Constant-Weight Codes

TL;DR

This paper extends the theory of diameter perfect codes to non-binary constant-weight codes under the Hamming metric, characterizing their structure, maximum anticodes, and connections to combinatorial theorems.

Contribution

It proves the code-anticode bound for non-binary constant-weight codes and classifies the families of diameter perfect codes and maximum size anticodes.

Findings

Six families of non-binary diameter constant-weight codes identified

Four families of maximum size anticodes characterized

Connections established with Erdős-Ko-Rado theorem

Abstract

Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weight anticodes. Each one of these families of diameter perfect codes raises some different questions. We consider some of these questions and leave lot of ground for further research. Finally, as a consequence, some t-intersecting families related to the well-known Erd\"{o}s-Ko-Rado theorem, are constructed.