Z4-Linear Perfect Codes

For every n = 2 k ≥ 16 there exist exactly ⌊(k + 1)/2⌋ mutually nonequiv-alent Z 4-linear extended perfect codes with distance 4. All these codes have different ranks. Codes represented in such a manner are called Z 4-linear. In [5] it is shown that the extended Golay code and the extended Hamming (n, 2 n−log 2 n−1 , 4)-codes (of length n and cardinality 2 n−log 2 n−1 , with distance 4) for every n > 16 are not Z 4-linear. Also, in [5] for every n = 2 k a Z 4-linear (2, 2 n−log 2 n−1 , 4)-code is described (the codes C 0,r 2 , in the notations of § 2, are presented as cyclic codes in [5]). The goal of this work is a complete description of Z 4-linear perfect and extended perfect codes. It is known [23, 22] that there are no nontrivial perfect binary codes except the Golay (23, 2 12 , 7)-code and the (2 k −1, 2 2 k −k−1 , 3)-codes. The perfect (23, 2 12 , 7)-code is unique up to equivalence. The linear (Hamming) (2 k − 1, 2 2 k −k−1 , 3)-code is also unique for every k, but for n = 2 k − 1 ≥ 15 there exist more than 2 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 k − 1, 2 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 k , 2 2 k −k−1 , 4)-codes can be represented as linear codes over the ring Z 4. In § 2, in terms of check matrices, we define ⌊(log 2 n + 1)/2⌋ Z 4-linear extended perfect (n, 2 n /2n, 4)-codes. In § 3 we show that the codes constructed are * Original Russian text was published in Diskretn.