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

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.