Trace-Function on a Galois Ring in Coding Theory
AAECC-12 Proceedings of the 12th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
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A classical binary Preparata code P2 (m) is a nonlinear (2m+1, 22(2m-1-m), 6)-code, where m is odd. It has a linear representation over the ring Z4 [Hammons et al., The Z4-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301-319]. Here for any q = 2l 2 and any m such that (m, q - 1 ) = 1 a nonlinear code Pq (m) over the field F = GF(q) with parameters (q(Δ + 1), q2(Δ-m), d ≥ 3q), where Δ=(qm-1)/(q - 1), is constructed. If d = 3q this set of parameters generalizes that of P2 (m). The equality d = 3q is established in the following cases: (1) for a series of initial admissible values q and m such that qm 100; (2) for m = 3, 4 and any admissible q, and (3) for admissible q and m such that there exists a number m1 with m1 |m and d(Pq (m1)) = 3q. We apply the approach of [Nechaev and Kuzmin, Linearly presentable codes, Proceedings of the 1996 IEEE International Symposium Information Theory and Application Victoria, BC, Canada 1996, pp. 31-34] the code P is a Reed-Solomon representation of a linear over the Galois ring R = GR(q2, 4) code P dual to a linear code K with parameters near to those of generalized linear Kerdock code over R.