# American Institute of Mathematical Sciences

May  2011, 5(2): 191-198. doi: 10.3934/amc.2011.5.191

## Some connections between self-dual codes, combinatorial designs and secret sharing schemes

 1 Department of Mathematics and Informatics, Veliko Tarnovo University, Bulgaria, Bulgaria

Received  March 2010 Revised  July 2010 Published  May 2011

In the present work we study a class of singly-even self-dual codes with the special property that the minimum weight of their shadow is 1. Some of these codes support 1 and 2-designs. Using them, we describe two types of schemes based on codes, the first is an one-part secret sharing scheme and the second is a two-part sharing scheme. Similar schemes can be constructed from self-dual codes that support 3-designs.
Citation: Stefka Bouyuklieva, Zlatko Varbanov. Some connections between self-dual codes, combinatorial designs and secret sharing schemes. Advances in Mathematics of Communications, 2011, 5 (2) : 191-198. doi: 10.3934/amc.2011.5.191
##### References:
 [1] E. F. Assmus and H. F. Mattson, New $5$-designs, J. Combin. Theory, 6 (1969), 122-151. doi: 10.1016/S0021-9800(69)80115-8. [2] S. Bouyuklieva and M. Harada, Extremal self-dual $[50,25,10]$ codes with automorphisms of order $3$ and quasi-symmetric $2-(49,9,6)$ designs, Des. Codes Crypt., 28 (2003), 163-169. doi: 10.1023/A:1022588407585. [3] J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931. [4] S. T. Dougherty, S. Mesnager and P. Solé, Secret-sharing schemes based on self-dual codes, in "Information Theory Workshop,'' Porto, (2008), 338-342. [5] W.C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl., 11 (2005), 451-490. doi: 10.1016/j.ffa.2005.05.012. [6] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge Univ. Press, 2003. [7] J. L. Massey, Some applications of coding theory in cryptography, in "Codes and Ciphers, Cryptography and Coding IV'' (ed. P.G. Farrell), Formara Lt, Esses, England, (1995), 33-47. [8] E. M. Rains, Shadow bounds for self-dual-codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.

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##### References:
 [1] E. F. Assmus and H. F. Mattson, New $5$-designs, J. Combin. Theory, 6 (1969), 122-151. doi: 10.1016/S0021-9800(69)80115-8. [2] S. Bouyuklieva and M. Harada, Extremal self-dual $[50,25,10]$ codes with automorphisms of order $3$ and quasi-symmetric $2-(49,9,6)$ designs, Des. Codes Crypt., 28 (2003), 163-169. doi: 10.1023/A:1022588407585. [3] J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931. [4] S. T. Dougherty, S. Mesnager and P. Solé, Secret-sharing schemes based on self-dual codes, in "Information Theory Workshop,'' Porto, (2008), 338-342. [5] W.C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl., 11 (2005), 451-490. doi: 10.1016/j.ffa.2005.05.012. [6] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge Univ. Press, 2003. [7] J. L. Massey, Some applications of coding theory in cryptography, in "Codes and Ciphers, Cryptography and Coding IV'' (ed. P.G. Farrell), Formara Lt, Esses, England, (1995), 33-47. [8] E. M. Rains, Shadow bounds for self-dual-codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.
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