A new distance-regular graph of diameter 3 on 1024 vertices

TL;DR

This paper constructs a new distance-regular graph of diameter 3 on 1024 vertices from a punctured dodecacode, and explores related strongly regular and completely regular codes and graphs.

Contribution

It introduces a novel distance-regular graph with a specific intersection array derived from a punctured dodecacode, and analyzes its automorphism group and related structures.

Findings

New distance-regular graph with intersection array {33,30,15;1,2,15}

Construction of a strongly regular graph with parameters (1024,495,238,240)

Development of a non-trivial completely regular binary code of length 33

Abstract

The dodecacode is a nonlinear additive quaternary code of length $12$. By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance $5$. In particular, this latter code is completely regular but not completely transitive. Its coset graph is distance-regular of diameter three on $2^{10}$ vertices, with new intersection array $\{33,30,15;1,2,15\}$. The automorphism groups of the code, and of the graph, are determined. Connecting the vertices at distance two gives a strongly regular graph of (previously known) parameters $(2^{10},495,238,240)$. Another strongly regular graph with the same parameters is constructed on the codewords of the dual code. A non trivial completely regular binary code of length $33$ is constructed.