An upper bound for the number of uniformly packed codes

Binary uniformly packed in the narrow sense codes were introduced in 1971 by Semakov, Zinoviev and Zaitsev. Later more general definitions were proposed by Bassalygo, Zaitsev and Zinoviev (uniformly packed in the wide sense codes) and by Goethals and Tilborg (uniformly packed codes). We consider binary uniformly packed in the wide sense codes. These codes are well known for their remarkable properties and have been intensively studied. In this paper we give an upper bound on the number of distinct uniformly packed in the wide sense codes of length n with constant odd minimum distance d and fixed parameters of packing. In particular, we give nontrivial upper bounds on the numbers of Preparata codes with d = 5, primitive BCH codes with d equal to 5 or 7, Goethals codes with d = 7, et al. The result obtained generalizes the upper bound for the number of perfect codes with d = 3 that was derived by Avgustinovich in 1995.