2024/11/14 by Goryainov, Sergey, Krotov, Denis
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.09698
In Cayley graphs on the additive group of a small vector space over GF(q), q=2,3, we look for completely regular (CR) codes whose parameters are new in Hamming graphs over the same field. The existence of a CR code in such Cayley graph G implies the existence of a CR code with the same parameters in the corresponding Hamming graph that covers G. In such a way, we find several completely regular codes with new parameters in Hamming graphs over GF(3). The most interesting findings are two new CR-1 (with covering radius~1) codes that are independent sets (such CR are equivalent to optimal orthogonal arrays attaining the Bierbrauer--Friedman bound) and one new CR-2. By recursive constructions, every knew CR code induces an infinite sequence of CR codes (in particular, optimal orthogonal arrays if the original code was CR-1 and independent). In between, we classify feasible parameters of CR codes in several strongly regular graphs.