On the Construction of Transitive Codes

Abstract

Application of some known methods of code construction (such as the Vasil'ev, Plotkin, and Mollard methods) to transitive codes satisfying certain auxiliary conditions yields infinite classes of large-length transitive codes, in particular, at least k/2 2 nonequivalent perfect transitive codes of length n = 2 k − 1, k > 4. A similar result is valid for extended perfect transitive codes.

Transitive nonpropelinear perfect codes

“…Now consider the Mollard code M (C, D). For a permutation π on the coordinate positions of the code C, denote by D 1 (π) a permutation on the coordinates of M(C, D): D 1 (π)(r, s) = (π(r), s) for r 0 and D 1 (π)(0, s) = (0, s) (see [23], [6]). For a permutation π on the coordinate positions of D, define D 2 (π)(r, s) = (r, π(s)) for s 0 and D 2 (π)(r, 0) = (r, 0).…”

On separability of the classes of homogeneous and transitive perfect binary codes

“…Using duplicator permutations for Rot(C), one can easily prove the following statement. Lemma 1 [9]. Let V λ C be the code obtained from a code C of length n using the Vasil'ev construction with the zero function λ. Then…”

On perfect binary codes

“…There are some other new results concerning automorphism groups of perfect codes, see [20,21,52,76]. In [78] applying some well-known constructions (Vasil'ev's, Plotkin's and Mollard's) to known binary transitive codes of some lengths and using some additional conditions infinite classes of transitive binary codes (not necessarily perfect codes) of greater lengths are presented. In particular case a class of perfect and extended perfect transitive codes for any admissible length n 31 is constructed.…”