The punctured dodecacode is unique

TL;DR

This paper proves the uniqueness of the punctured dodecacode, a special additive code, by showing its properties, classifying related codes, and establishing nonexistence results in certain graph structures.

Contribution

It demonstrates the uniqueness of the punctured dodecacode, classifies related two-weight codes, and proves the nonexistence of similar codes in Doob graphs.

Findings

The punctured dodecacode is unique among codes with its weight distribution.

It is shown to be nonlinear.

No analogues of the dodecacode exist in Doob graphs.

Abstract

The punctured dodecacode is an additive $4$-ary code of length $11$ and distance $5$ which is uniformly packed. We show that a code with the same weight distribution is equivalent to it. This code is also shown to be nonlinear. We also establish the nonexistence of analogues of the dodecacode and the punctured dodecacode in Doob graphs. To that end, we classify two-weight codes of weights $6$ and $8$ in Doob and $4$-ary Hamming graphs of diameter $9$ and the corresponding strongly regular graphs. Keywords: dodecacode, additive code, trace Hermitian duality, uniformly packed code, completely regular code, Doob graph, strongly regular graph