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On optimal superimposed codes. (English) Zbl 1051.94013
A family of subsets of a finite set is called $$(w,r)$$ cover-free if no intersection of $$w$$ members of the family is contained in the union of $$r$$ others [cf. D. R. Stinson, L. Zhu and R. Wei, J. Comb. Theory, Ser. A 90, 224–234 (2000; Zbl 0948.05055) and A. D’yachkov, A. Macula, D. Torney and P. Vilenkin, ibid. 99, 195–218 (2002; Zbl 1020.94027)]. Equivalent concepts are that of (binary) $$(w,r)$$ superimposed codes (they are given by the incidence matrices of $$(w,r)$$ cover-free families) and of $$(w,r)$$ key distribution patterns for cryptography.
The authors develop methods of constructing $$(w,r)$$ superimposed codes using combinatorial designs and 3-covering arrays and prove that some of the $$(2,2)$$ and ($$2,3)$$ superimposed codes found by their methods are optimal. To get good superimposed codes of large size, they concatenate algebraic-geometric codes with a $$(w,r)$$ superimposed code; thus they obtain a sequence of $$(w,r)$$ superimposed codes of positive asymptotic rate.

##### MSC:
 94B25 Combinatorial codes 05D05 Extremal set theory 94A60 Cryptography 05B05 Combinatorial aspects of block designs 05B40 Combinatorial aspects of packing and covering
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##### References:
 [1] Adams, Aequationes Mathematicae 51 pp 230– (1995) [2] Alon, Discrete Math 58 pp 191– (1986) [3] Chateauneuf, J Combin Designs 10 pp 217– (2002) [4] Chen, J Statistical Planning Infer 51 pp 339– (1996) [5] Chen, J Combin Math Combin Comput 17 pp 149– (1995) [6] Chor, IEEE Transactions Inform Theory 46 pp 893– (2000) [7] Cohen, IEEE Transactions Infor Theory 40 pp 1872– (1994) [8] and Intersecting codes and identifying codes, Proc Intern Workshop Coding Cryptography, January 8-12, 2001, Paris (France), pp. 139-147. [9] and CRC Handbook Combin Designs CRC Press, Inc., 1996. [10] and Combinatorial group testing and its applications, World Scientific, Singapore, New Jersey, London, Hong Kong, 1993. · Zbl 0867.90060 [11] D’yachkov, Problemy Peredachi Informatsii 18 pp 7– (1982) [12] D’yachkov, Problems Control Inform Theory 12 pp 229– (1983) [13] D’yachkov, Problems Control Inform Theory 18 pp 237– (1989) [14] D’yachkov, IEEE Trans Inform Theory 46 (2000) [15] and New applications and results of superimposed code theory arising from the potentialities of molecular biology, Numbers, Inform Complexity, Kluwer Academic Publishers, 2000, pp. 265-282. · Zbl 1034.94015 [16] and New results in the theory of superimposed codes, Seventh International Workshop Algebraic Combin Coding Theory, June 18-24, 2000, Bansko (Bulgaria), pp. 126-136. [17] D’yachkov, J Combin Theory, Series A 99 pp 195– (2002) [18] Dyer, J Cryptol 8 pp 189– (1995) [19] Erdos, J Combin Theory 33 pp 158– (1982) [20] Erdos, Israel J Math 51 pp 75– (1985) [21] Gronau, J Combin Math Combin Comput 11 pp 113– (1992) [22] Kautz, IEEE Trans Inform Theory IT-10 pp 363– (1964) [23] Korner, J Combin Theory, Series A 71 pp 112– (1995) [24] and The Theory of error-correcting codes, North Holland, 1983. [25] Mago, IEEE Trans Comput 22 pp 928– (1973) [26] Mitchell, Discrete Applied Math 21 pp 215– (1988) [27] O’Keefe, Designs Codes Cryptography 5 pp 261– (1995) [28] Quinn, Designs, Codes Cryptography 4 pp 177– (1994) [29] Quinn, J Cryptology 12 pp 227– (1999) [30] Sagalovich, Probl Inform Transmission 30 pp 105– (1994) [31] Sagalovich, Problemy Peredachi Informatsii 18 pp 74– (1982) [32] Sloane, J Combin Designs 1 pp 51– (1993) [33] Stinson, Designs, Codes Cryptography 12 pp 215– (1997) [34] Stinson, J Combin Designs 8 pp 189– (2000) [35] Stinson, J Combin Theory Ser A 90 pp 224– (2000) [36] Stinson, key distribution patterns, group testing algorithms and related structures, J Stat Planning Infer 86 pp 595– (2000) · Zbl 1054.94013 [37] Tsfasmann, Discrete Applied Math 33 pp 241– (1991) [38] Wang, J Combin Theory Ser A 93 pp 112– (2001)
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