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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

Stefka Bouyuklieva 1, and  Zlatko Varbanov 1,

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.
Keywords: Self-dual codes, secret sharing schemes, combinatorial designs..
Mathematics Subject Classification: Primary: 94B05, 94A62; Secondary: 05B0.
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.  Google Scholar

[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.  Google Scholar

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S. T. Dougherty, S. Mesnager and P. Solé, Secret-sharing schemes based on self-dual codes, in "Information Theory Workshop,'' Porto, (2008), 338-342. Google Scholar

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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.  Google Scholar

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W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge Univ. Press, 2003.  Google Scholar

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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. Google Scholar

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E. M. Rains, Shadow bounds for self-dual-codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge Univ. Press, 2003.  Google Scholar

[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. Google Scholar

[8]

E. M. Rains, Shadow bounds for self-dual-codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.  Google Scholar

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