# Representation of ℤ4-Linear Preparata Codes Using Vector Fields

Representation of ℤ4-Linear Preparata Codes Using Vector Fields 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# Representation of ℤ4-Linear Preparata Codes Using Vector Fields

, Volume 41 (2) – Jul 15, 2005
12 pages
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Publisher
Nauka/Interperiodica
Copyright
Copyright © 2005 by MAIK “Nauka/Interperiodica”
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1007/s11122-005-0016-4
Publisher site
See Article on Publisher Site

### 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.

### Journal

Problems of Information TransmissionSpringer Journals

Published: Jul 15, 2005

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