A generalization of the binary Preparata code

A classical binary Preparata code P2(m)P2(m) is a nonlinear (2m+1,22(2m-1-m),6)(2m+1,22(2m-1-m),6)-code, where m   is odd. It has a linear representation over the ring Z4Z4 [Hammons et al., The Z4Z4-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301–319]. Here for any q=2l>2q=2l>2 and any m   such that (m,q-1)=1(m,q-1)=1 a nonlinear code Pq(m)Pq(m) over the field F=GF(q)F=GF(q) with parameters (q(Δ+1),q2(Δ-m),d⩾3q)(q(Δ+1),q2(Δ-m),d⩾3q), where Δ=(qm-1)/(q-1)Δ=(qm-1)/(q-1), is constructed. If d=3qd=3q this set of parameters generalizes that of P2(m)P2(m). The equality d=3qd=3q is established in the following cases: (1) for a series of initial admissible values q and m   such that qm<2100qm<2100; (2) for m=3,4m=3,4 and any admissible q, and (3) for admissible q and m   such that there exists a number m1m1 with m1|mm1|m and d(Pq(m1))=3qd(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)R=GR(q2,4) code PP dual to a linear code KK with parameters near to those of generalized linear Kerdock code over R.