, Volume 53, Issue 2, pp 217-230

On mappings of the Galois space

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Abstract

LetV be a metric vector space over a fieldK, dimV=n<∞, and let δ:V×VK denote the corresponding distance function. Given a mappingσ:VV such that δ(p,q) = 1⇒ δ(p σ ,q ς) = 1, ifn=2, indV=1 and charK≠2, 3, 5, thenσ is semilinear [5], [11]; ifn≧3,K=R and the distance function is either Euclidean or Minkowskian, thenσ is linear [3], [10]. Here the following is proved: IfK=GF(p m ),p>2 andn≧3, thenσ is semilinear (up to a translation), providedn≠0, −1, −2 (modp) or the discriminant ofV satisfies a certain condition. The proof is based on the condition for a regular simplex to exist in a Galois space, which may be of interest for its own sake.