Geodesic
Binary representations of underdetermined data and superimposed codes
Prikladnaâ diskretnaâ matematika, no. 1 (2013), pp. 17-33.

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For underdetermined data, binary representations are defined, making it possible to reconstruct the initial data entirely (not only their specifications) and being fairly compact. To solve the problem of their design, some special matrices called selective ones are introduced and studied. They generalize the disjunct (cover-free) matrices widely applied in Computer Science. Some characteristics of selective matrices and estimates of data representation length via some data parameters are investigated. Problems related to the complexity of representations design are considered too.
Keywords: underdetermined data, binary representation, sets system basis, representation length, disjunct matrix, superimposed code, cover-free family
Mots-clés : compression, polynomial algorithm. 
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L. A. Sholomov. Binary representations of underdetermined data and superimposed codes. Prikladnaâ diskretnaâ matematika, no. 1 (2013), pp. 17-33. http://geodesic.mathdoc.fr/item/PDM_2013_1_a2/

[1] Sholomov L. A., “Elementy teorii nedoopredelennoi informatsii”, Prikladnaya diskretnaya matematika. Prilozhenie, 2009, no. 2, 18–42

[2] Sholomov L. A., “Dvoichnoe predstavlenie nedoopredelennykh dannykh”, Doklady Akademii nauk, 448:3 (2013), 275–278 | DOI

[3] Kautz W. H., Singleton R. C., “Nonrandom binary superimposed codes”, IEEE Trans. Inform. Theory, 10:4 (1964), 363–377 | DOI | Zbl

[4] Dyachkov A. G., Rykov V. V., “Granitsy dliny diz'yunktivnykh kodov”, Problemy peredachi informatsii, 18:3 (1982), 7–13 | MR

[5] Kumar R., Rajagopalan S., Sahai A., “Coding construction for blacklisting problems without computational assumptions”, LNCS, 1666, 1999, 609–623 | Zbl

[6] Porat E. and Rotshchild A., “Explicit non-adaptive combinatorial group testing schemes”, Automata, Languages and Programming, LNCS, 5125, 2008, 748–759 | MR | Zbl

[7] Chuzhoy J. and Khanna S., “An O$(k^3\log n)$-approximation algorithm for vertex-connectivity survivable network design”, FOCS, Proc. 50th Annual IEEE Symp. Foundation Computer Science, IEEE Computer Society, Washington, 2009, 437–441 | MR

[8] Dyer M., Fenner T., Frieze A., Thomason A., “On key storage in secure networks”, J. Cryptology, 8 (1995), 189–200 | DOI | Zbl

[9] Mitchell C. J., Piper F. C., “Key storage in secure networks”, Discr. Appl. Math., 21 (1988), 215–228 | DOI | MR | Zbl

[10] Quinn K. A. S., “Bounds for key distribution patterns”, J. Cryptology, 12 (1999), 227–239 | DOI | Zbl

[11] Geri M., Dzhonson D., Vychislitelnye mashiny i trudnoreshaemye zadachi, Mir, M., 1982, 416 pp.

[12] Kospanov E. Sh., “O kodirovanii (0,1)-matrits kon'yunktsiyami”, Sbornik trudov, Diskretnyi analiz, 27, Institut matematiki SO AN, Novosibirsk, 1975, 17–22

[13] Erdos P., Frankl P., Furedi Z., “Families of finite sets in which no set is covered by the union of $r$ others”, Israel J. Math., 51:1–2 (1985), 79–89 | DOI | MR

[14] Furedi Z., “On $r$-cover-free families”, J. Combinator. Theory. Ser. A, 73 (1996), 172–173 | DOI | MR

[15] Cheng V., Du D.-Z., Lin G., “On the upper bounds of the minimum number of rows of disjunct matrices”, Optimizat. Lett., 3:2 (2009), 297–302 | DOI | MR

[16] Hwang F. K., Sos V. T., “Non-adaptive hypergeometric group testing”, Studia Scientiarum Mathecarum Hungarica, 22, 1987, 257–263 | MR | Zbl

[17] Ruszinko M., “On the upper bound of the size of the $r$-cover-free families”, J. Combinator. Theory. Ser. A, 66 (1994), 302–310 | DOI | MR | Zbl

[18] D'yachkov A. G., Macula A. J., Rykov V. V., “New constructions of superimposed codes”, IEEE Trans. Inform. Theory, 46:1 (2000), 284–290 | DOI | MR