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Representation of ℤ4-Linear Preparata Codes Using Vector Fields

  • Coding Theory
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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 \(\mathcal{H}_{\lambda ,\not \upsilon } + \mathcal{M}\) with

$$\mathcal{H}_{\lambda ,\not \upsilon } = \{ y + T_\lambda (y) + S_{\not \upsilon } (y)|y \in H^n \} ,\quad \mathcal{M} = 2H^n ,$$

where T λ(⋅) and S ψ (⋅) are vector fields of a special form defined over the binary extended linear Hamming code H n of length n. An upper bound on the number of nonequivalent quaternary linear Preparata codes of length n is obtained, namely, \(2^{n\log _2 n}\). A representation for binary Preparata codes contained in perfect Vasil’ev codes is suggested.

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Authors and Affiliations

  1. Novosibirsk State University, Russia

    N. N. Tokareva

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__________

Translated from Problemy Peredachi Informatsii, No. 2, 2005, pp. 50–62.

Original Russian Text Copyright © 2005 by Tokareva.

Supported in part by the Ministry of Education of the Russian Federation program “Development of the Scientific Potential of the Higher School,” project no. 512.

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Tokareva, N.N. Representation of ℤ4-Linear Preparata Codes Using Vector Fields. Probl Inf Transm 41, 113–124 (2005). https://doi.org/10.1007/s11122-005-0016-4

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  • DOI: https://doi.org/10.1007/s11122-005-0016-4

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