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Diskr. Mat., 1998, Volume 10, Issue 2, Pages 3–29 (Mi dm424)  

This article is cited in 11 scientific papers (total in 11 papers)

Recursive MDS-codes and recursively differentiable quasigroups

S. González, E. Couselo, V. T. Markov, A. A. Nechaev


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.

DOI: https://doi.org/10.4213/dm424

Full text: PDF file (2375 kB)

English version:
Discrete Mathematics and Applications, 1998, 8:3, 217–245

Bibliographic databases:

UDC: 519.7
Received: 10.03.1998

Citation: S. González, E. Couselo, V. T. Markov, A. A. Nechaev, “Recursive MDS-codes and recursively differentiable quasigroups”, Diskr. Mat., 10:2 (1998), 3–29; Discrete Math. Appl., 8:3 (1998), 217–245

Citation in format AMSBIB
\Bibitem{GonCouMar98}
\by S.~Gonz\'alez, E.~Couselo, V.~T.~Markov, A.~A.~Nechaev
\paper Recursive MDS-codes and recursively differentiable quasigroups
\jour Diskr. Mat.
\yr 1998
\vol 10
\issue 2
\pages 3--29
\mathnet{http://mi.mathnet.ru/dm424}
\crossref{https://doi.org/10.4213/dm424}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1673150}
\zmath{https://zbmath.org/?q=an:0982.94028}
\transl
\jour Discrete Math. Appl.
\yr 1998
\vol 8
\issue 3
\pages 217--245


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Couselo E., Gonzalez S., Markov V., Nechaev A., “Recursive MDS-codes and pseudogeometries”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Proceedings, Lecture Notes in Computer Science, 1719, 1999, 211–220  crossref  mathscinet  zmath  isi  scopus
    2. Kurakin V.L., Kuzmin A.S., Markov V.T., Mikhalev A.V., Nechaev A.A., “Linear codes and polylinear recurrences over finite rings and modules (a survey)”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Proceedings, Lecture Notes in Computer Science, 1719, 1999, 365–391  crossref  mathscinet  zmath  isi  scopus
    3. A. S. Abashin, “Linear recursive MDS-codes of dimensions 2 and 3”, Discrete Math. Appl., 10:3 (2000), 319–332  mathnet  crossref  mathscinet  zmath
    4. S. González, E. Couselo, V. Markov, A. Nechaev, “The parameters of recursive MDS-codes”, Discrete Math. Appl., 10:5 (2000), 433–453  mathnet  crossref  mathscinet  zmath
    5. Nechaev A.A., “Recurring sequences”, Formal Power Series and Algebraic Combinatorics, 2000, 54–66  crossref  mathscinet  zmath  isi
    6. S. González, E. Couselo, V. T. Markov, A. A. Nechaev, “Group codes and their nonassociative generalizations”, Discrete Math. Appl., 14:2 (2004), 163–172  mathnet  crossref  crossref  mathscinet  zmath
    7. 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  mathnet  crossref  mathscinet
    8. V. T. Markov, A. V. Mikhalev, A. V. Gribov, P. A. Zolotykh, S. S. Skazhenik, “Kvazigruppy i koltsa v kodirovanii i postroenii kriptoskhem”, PDM, 2012, no. 4(18), 31–52  mathnet
    9. Iryna Fryz, “Orthogonality and retract orthogonality of operations”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2018, no. 1, 24–33  mathnet
    10. Fryz I.V., “Algorithm For the Complement of Orthogonal Operations”, Comment. Math. Univ. Carol., 59:2 (2018), 135–151  crossref  mathscinet  zmath  isi  scopus
    11. 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  mathnet  crossref  crossref
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