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

Minjia Shi (1) , Denis S. Krotov (2) , Patrick Sole (3)

1 Anhui University [Hefei]
2 Sobolev Institute of Mathematics
3 LAGA - Laboratoire Analyse, Géométrie et Applications

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.

Domaines: Combinatoire [math.CO]

Minjia Shi, Denis S. Krotov, Patrick Sole. A new distance-regular graph of diameter 3 on 1024 vertices. Designs, Codes and Cryptography, 2019, 87, pp.2091 - 2101. ⟨10.1007/s10623-019-00609-w⟩. ⟨hal-02411616⟩