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There is a strongly regular graph Γ with parameters (v, k, λ, μ) = (1288, 792, 476, 504). The spectrum is 7921 81035 (–36)252. Its complement has parameters (v, k, λ, μ) = (1288, 495, 206, 180) and spectrum 4951 35252 (–9)1035. Construction is folklore. The graph is for example given in Hubaut (1975). Uniqueness is unknown.

## Construction

Let C be the extended binary Golay code. It has 2576 words of weight 12 (dodecads), so 1288 complementary pairs of dodecads. Given one dodecad, there are 1, 495, 1584, 495, 1 dodecads at distance 0, 8, 12, 16, 24, respectively. Given one complementary pair of dodecads, there are 1, 495, 792 such pairs at distance 0, 8, 12, respectively. The graph Γ is obtained if we call two dodecad pairs adjacent if they have distance 12.

## Group

The automorphism group of Γ is M24 acting rank 3 with point stabilizer M12:2.

## Supergraphs

This graph is the local graph of a strongly regular graph Σ with parameters (v, k, λ, μ) = (2048, 1288, 792, 840) and spectrum 12881 81771 (–56)276. Its complement has parameters (v, k, λ, μ) = (2048, 759, 310, 264) and spectrum 7591 55276 (–9)1771. The graph Σ was constructed by Goethals and Seidel (1970), who write `we have reasons to believe that the subgraph on the 1288 neighbours of any vertex is strongly regular'. Uniqueness is unknown.

## Construction

Let C be the extended binary Golay code. It has 1, 759, 2576, 759, 1 words of weight 0, 8, 12, 16, 24, respectively. Take the 211 cosets of {0,1} in C, and join two cosets when they have distance 12.

## Group

The automorphism group of Σ is 211.M24 acting rank 3 with point stabilizer M24.

## References

J. M. Goethals & J. J. Seidel, Strongly regular graphs derived from combinatorial designs, Canad. J. Math. 22 (1970) 597-614.

X. L. Hubaut, Strongly regular graphs, Discr. Math. 13 (1975) 357-381.