Non-existence of a ternary constant weight $(16, 5, 15; 2048)$ diameter perfect code

TL;DR

This paper proves the non-existence of certain ternary constant weight codes with specific parameters for m=4 and establishes that all such codes are diameter perfect, contributing to the understanding of code existence and structure.

Contribution

The paper demonstrates the non-existence of ternary constant weight codes with given parameters for m=4 and shows these codes are diameter perfect, filling gaps in coding theory knowledge.

Findings

Non-existence of codes for m=4

Codes are diameter perfect when they exist

Existence depends on APN permutation conditions

Abstract

Ternary constant weight codes of length $n=2^m$, weight $n-1$, cardinality $2^n$ and distance $5$ are known to exist for every $m$ for which there exists an APN permutation of order $2^m$, that is, at least for all odd $m \geq 3$ and for $m=6$. We show the non-existence of such codes for $m=4$ and prove that any codes with the parameters above are diameter perfect.