CODING THEORY Representation of Z4-Linear Preparata Codes Using Vector Fields1

Abstract—A binary code is called Z4-linear if its quaternary Gray map preimage is linear. We show that the set of all quaternary linear Preparata codes of length n = 2m, m odd, m ≥ 3, is nothing more than the set of codes of the form Hλ,ψ +M with Hλ,ψ = {y + Tλ(y) + Sψ(y) | y ∈ Hn}, M = 2Hn, where Tλ(·) and Sψ(·) are vector fields of a special form defined over the binary extended linear Hamming code Hn of length n. An upper bound on the number of nonequivalent quaternary linear Preparata codes of length n is obtained, namely, 2n log2 n. A representation for binary Preparata codes contained in perfect Vasil’ev codes is suggested. 1