The parameters of recursive MDS-codes

S. González ; E. Couselo ; V. Markov ; A. Nechaev

Diskretnaya Matematika, Tome 12 (2000) no. 4, pp. 3-24

Résumé

A full $m$-recursive code of length $n>m$ over an alphabet of $q\geq 2$ elements is the set of all segments of length $n$ of the recurring sequences that satisfy some fixed recursivity law $f(x_1,\dots,x_m)$. We investigate the conditions under which there exist such codes with distance $n-m+1$ (recursive MDS-codes). Let $\nu^r(m,q)$ be the maximum of the numbers $n$ for which a full $m$-recursive code exists. In our previous paper, it was noted that the condition $\nu^r(m,q)\geq n$ means that there exists an $m$-quasigroup $f$, which together with its $n-m-1$ sequential recursive derivatives forms an orthogonal system of $m$-quasigroups (of Latin squares for $m=2$). It was proved that $\nu^r(m,q)\geq 4$ for all values $q\in\mathbf N$ except possibly six of them. Here we strengthen this estimate for a series of values $q100$ and give some lower bounds for $\nu^r(m,q)$ for $m>2$. In particular, we prove that $\nu^r(m, q) \ge q+1$ for all primary $q$ and $m=1,\dots,q$ and $\nu^r(2^t-1,2^t)=2^t+2$ for $t = 2,3,4$. Moreover, we prove that there exists a linear recursive $[6,3,4]$-MDS-code over the group $Z_2\oplus Z_2$, but there is no such code over the field $F_4$.