1 O ct 2 00 7 Z 4 - Linear Perfect Codes *

Abstract

For every n = 2k ≥ 16 there exist exactly ⌊(k + 1)/2⌋ mutually nonequivalent Z4-linear extended perfect codes with distance 4. All these codes have different ranks. Certain of known nonlinear binary codes such as Kerdock, Preparata, Goethals, Delsarte-Goethals codes can be represented, using some mapping Z4 → Z 2 2 (in this paper, following [5], we use the mapping 0 → 00, 1 → 01, 2 → 11, 3 → 10) as linear codes over the alphabet {0, 1, 2, 3} with modulo 4 operations (see [14, 10, 11, 12, 5]). Codes represented in such a manner are called Z4-linear. In [5] it is shown that the extended Golay code and the extended Hamming (n, 22 , 4)-codes (of length n and cardinality 22 , with distance 4) for every n > 16 are not Z4-linear. Also, in [5] for every n = 2 a Z4-linear (2, 2 n−log2 , 4)-code is described (the codes C2, in the notations of § 2, are presented as cyclic codes in [5]). The goal of this work is a complete description of Z4-linear perfect and extended perfect codes. It is known [23, 22] that there are no nontrivial perfect binary codes except the Golay (23, 2, 7)-code and the (2−1, 2 k−k−1, 3)-codes. The perfect (23, 2, 7)-code is unique up to equivalence. The linear (Hamming) (2 − 1, 2 k−k−1, 3)-code is also unique for every k, but for n = 2 − 1 ≥ 15 there exist more than 2 (n+1)/2−k (for the last lower bound, see [6]) nonlinear codes with the same parameters (see, e.g., [19, 3] for a survey of some constructions). The class of all (2 − 1, 2 k−k−1, 3)-codes is not described yet. In this paper we show that not great, but increasing as k → ∞, number of extended perfect (2, 2 k−k−1, 4)-codes can be represented as linear codes over the ring Z4. In § 2, in terms of check matrices, we define ⌊(log2 n + 1)/2⌋ Z4-linear extended perfect (n, 2/2n, 4)-codes. In § 3 we show that the codes constructed are Original Russian text was published in Diskretn. Anal. Issled. Oper., Ser. 1, 7(4):78-90, 2000.

Cite this paper

@inproceedings{Krotov20071OC, title={1 O ct 2 00 7 Z 4 - Linear Perfect Codes *}, author={Denis S. Krotov}, year={2007} }