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. 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. 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. doi: 10.1109/18.59931.

[4]

S. T. Dougherty, S. Mesnager and P. Solé, Secret-sharing schemes based on self-dual codes,, in, (2008), 338.

[5]

W.C. Huffman, On the classification and enumeration of self-dual codes,, Finite Fields Appl., 11 (2005), 451. 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, (1995), 33.

[8]

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

show all references

References:
[1]

E. F. Assmus and H. F. Mattson, New $5$-designs,, J. Combin. Theory, 6 (1969), 122. 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. 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. doi: 10.1109/18.59931.

[4]

S. T. Dougherty, S. Mesnager and P. Solé, Secret-sharing schemes based on self-dual codes,, in, (2008), 338.

[5]

W.C. Huffman, On the classification and enumeration of self-dual codes,, Finite Fields Appl., 11 (2005), 451. 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, (1995), 33.

[8]

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

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