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5 papers
The parameters of recursive MDS-codes
S. González,
E. Couselo,
V. Markov,
A. Nechaev
Abstract:
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
$q<100$ 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
$F_4$.
UDC:
519.7 Received: 26.06.2000
DOI:
10.4213/dm353