Classification and Galois conjugacy of Hamming maps

Gareth A. Jones

Abstract


We show that for each d ≥ 1 the d-dimensional Hamming graph H(d, q) has an orientably regular surface embedding if and only if q is a prime power pe. If q > 2 there are up to isomorphism φ(q − 1)/e such maps, all constructed as Cayley maps for a d-dimensional vector space over the field Fq. We show that for each such pair (d, q) the corresponding Belyi pairs are conjugate under the action of the absolute Galois group Gal \overline Q, and we determine their minimal field of definition. We also classify the orientably regular embedding of merged Hamming graphs for q > 3.

Keywords


Hamming graph, Hamming map, automorphism group, Galois group.

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DOI: https://doi.org/10.26493/1855-3974.130.cba

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications