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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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