Perfect binary codes: constructions, properties, and enumeration

Abstract
Properties of nonlinear perfect binary codes are investigated and several new constructions of perfect codes are derived from these properties. An upper bound on the cardinality of the intersection of two perfect codes of length n is presented, and perfect codes whose intersection attains the upper bound are constructed for all n. As an immediate consequence of the proof of the upper bound the authors obtain a simple closed-form expression for the weight distribution of a perfect code. Furthermore, they prove that the characters of a perfect code satisfy certain constraints, and provide a sufficient condition for a binary code to be perfect. The latter result is employed to derive a generalization of the construction of Phelps (1983), which is shown to give rise to some perfect codes that are nonequivalent to the perfect codes obtained from the known constructions. Moreover, for any m⩾4 the authors construct full-rank perfect binary codes of length 2m -1. These codes are obviously nonequivalent to any of the previously known perfect codes. Furthermore the latter construction exhibits the existence of full-rank perfect tilings. Finally, they construct a set of 2(2cn) nonequivalent perfect codes of length n, for sufficiently large n and a constant c=0.5-ε. Precise enumeration of the number of codes in this set provides a slight improvement over the results reported by Phelps

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