Abstract:
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 $f(x_1,\ldots,x_k)$ 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 $q\notin\{2,6,14,18,26,42\}$, then $n^r(2,q)\geq 4$.
The proof is reduced to constructing some special pairs of orthogonal
Latin squares.