Bernstein superalgebras are introduced and irreducible Bernstein supermodules are classified.

Kerdock codes (Kerdock, Inform Control 20:182-187, 1972) are a wellknown family of non-linear binary codes with good parameters admitting a linear presentation in terms of codes over the ring Z 4 (see Nechaev, Diskret Mat 1:123-139, 1989; Hammons et al., IEEE Trans Inform Theory 40:301-319, 1994). These codes have been generalized in different directions: in Calderbank et al. (Proc Lond Math Soc 75:436-480, 1997) a symplectic construction of non-linear binary codes with the same parameters of the Kerdock codes has been given. Such codes are not necessarily equivalent. On the other hand, in Kuzmin and Nechaev (Russ Math Surv 49(5), 1994) the authors give a family of non-linear codes over the finite field F of q = 2 l elements, all of them admitting a linear presentation over the Galois Ring R of cardinality q 2 and characteristic 2 2 . The aim of this article is to merge both approaches, obtaining in this way new families of non-linear codes over F that can be presented as linear codes over the Galois Ring R. The construction uses symplectic spreads.

A code of length n over an alphabet of q > 2 elements is called a full ^-recursive code if it consists of all segments of length π of a recurring sequence that satisfies some fixed (nonlinear in general) recursivity law f(x\ , . . . , jty) of order k < n. Let n r (k, q) be the maximal number n such that there exists such a code with distance n -k+ 1 (MDS-code). The condition n r (k,q) > n means that the function / together with its n -k -1 sequential recursive derivatives forms an orthogonal system of fc-quasigroups. We prove that if q £ {2,6, 14, 18,26,42}, then n r (2,q) > 4. The proof is reduced to constructing some special pairs of orthogonal Latin squares.

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