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 Diskr. Mat., 2000, Volume 12, Issue 4, Pages 3–24 (Mi dm353)

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 $f(x_1,…,x_m)$. 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 $m=1,…,q$ 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 $F_4$.

DOI: https://doi.org/10.4213/dm353

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English version:
Discrete Mathematics and Applications, 2000, 10:5, 433–453

Bibliographic databases:

UDC: 519.7

Citation: S. González, E. Couselo, V. Markov, A. Nechaev, “The parameters of recursive MDS-codes”, Diskr. Mat., 12:4 (2000), 3–24; Discrete Math. Appl., 10:5 (2000), 433–453

Citation in format AMSBIB
\Bibitem{GonCouMar00} \by S.~Gonz\'alez, E.~Couselo, V.~Markov, A.~Nechaev \paper The parameters of recursive MDS-codes \jour Diskr. Mat. \yr 2000 \vol 12 \issue 4 \pages 3--24 \mathnet{http://mi.mathnet.ru/dm353} \crossref{https://doi.org/10.4213/dm353} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1826175} \zmath{https://zbmath.org/?q=an:1020.94020} \transl \jour Discrete Math. Appl. \yr 2000 \vol 10 \issue 5 \pages 433--453 

• http://mi.mathnet.ru/eng/dm353
• https://doi.org/10.4213/dm353
• http://mi.mathnet.ru/eng/dm/v12/i4/p3

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This publication is cited in the following articles:
1. V. T. Markov, A. A. Nechaev, S. Skazhenik, E. O. Tveritinov, “Pseudogeometries with clusters and an example of a recursive $[4,2,3]_{42}$-code”, J. Math. Sci., 163:5 (2009), 563–571
2. O. A. Kozlitin, “Veroyatnostnye lineinye sootnosheniya v dvoichnykh rekurrentnykh posledovatelnostyakh”, Matem. vopr. kriptogr., 8:3 (2017), 57–84
3. M. I. Rozhkov, S. S. Malakhov, “Experimental methods for constructing MDS matrices of a special form”, J. Appl. Industr. Math., 13:2 (2019), 302–309
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