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Recursive MDS-codes and recursive differentiable quasigroups. (English. Russian original) Zbl 0982.94028
Discrete Math. Appl. 8, No. 3, 217-245 (1998); translation from Diskretn. Mat. 10, No. 2, 3-29 (1998).
Summary: A code of length $$n$$ over an alphabet of $$q\geq 2$$ elements is called a full $$k$$-recursive code if it consists of all segments of length $$n$$ of a recurring sequence that satisfies some fixed (nonlinear in general) recursivity law $$f(x_1,\dots, x_k)$$ of order $$k\leq n$$. 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 $$n^r(k,q)\geq n$$ means that the function $$f$$ together with its $$n-k-1$$ sequential recursive derivatives forms an orthogonal system of $$k$$-quasigroups. We prove that if $$q\not\in \{2,6,14,18,26,42\}$$, then $$n^r(2,q)\geq 4$$. The proof is reduced to constructing some special pairs of orthogonal Latin squares.

##### MSC:
 94B60 Other types of codes 20N05 Loops, quasigroups 05B15 Orthogonal arrays, Latin squares, Room squares
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