A new distance-regular graph of diameter 3 on 1024 vertices
Minjia Shi, Denis Krotov, Patrick Sol\'e

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 . By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance . In particular, this latter code is completely regular but not completely transitive. Its coset graph is distance-regular of diameter three on vertices, with new intersection array . 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 . 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 is constructed.
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