# THE CLASSIFICATION OF SELF-DUAL CODES OF LENGTH 6 OVER ℤm FOR SMALL m

• Park, Young Ho (Department of Mathematics Kangwon National University)
• Accepted : 2010.12.13
• Published : 2010.12.30

#### Abstract

In this article we study self-dual codes of length 6 over ${\mathbb{Z}}_m$. A classification of such codes for $m{\leq}24$ is given. Main tool for the classification is the new double cosets decomposition method given in the recent article of the author.

#### References

1. J. M. P. Balmaceda, R. A. L. Betty and F. R. Nemenzo, Mass formula for self-dual codes over $Z_{p2}$ , Discrete Math. 308 (2009), 2984-3002.
2. K. Betsumiya, S. Georgiou, T. A. Gulliver, M. Harada, C. Kououvinos, On self-dual codes over some prime fields, Discrete Math. 262 (2009), 37-58.
3. J. H. Conway and N. J. A. Sloane, Self-dual codes over the integers modulo 4, J. Comin. Theory Ser. A 62 (1993), 30-45. https://doi.org/10.1016/0097-3165(93)90070-O
4. S. T. Dougherty, T. A. Gulliver and J. N. C. Wong, Self-dual codes over $Z_{8}$ and $Z_{9}$, Des Codes Crypt 41 (2006), 235-249. https://doi.org/10.1007/s10623-006-9000-2
5. S. T. Dougherty, M. Harada and P. Sole, Self-dual codes over rings and the Chinese Remainder Theorem, Hokkaido Math Journal 28 (1999), 253-283. https://doi.org/10.14492/hokmj/1351001213
6. S. T. Dougherty, S. Y. Kim and Y. H. Park, Lifted codes and their weight enumerators, Discrite Math. 305 (2005), 123-135. https://doi.org/10.1016/j.disc.2005.08.004
7. S. T. Dougherty and Y. H. Park, Codes over the p-adic integers, Des. Codes. Cryptogr. 39 (2006), 65-80. https://doi.org/10.1007/s10623-005-2542-x
8. M. Harada and A. Munemasa, On the classfication of self-dual Zk-codes, Lec- ture Notes in Computer Science 5921, (2009) 78-90.
9. W. C. Huffman, On the classification and enumeration of self-dual codes, Finite fields and their applications, 11 (2005), 451-490. https://doi.org/10.1016/j.ffa.2005.05.012
10. W. C. Huffman and V. Pless, Fundamentals of error-correcting codes, Cam- bridge, 2003.
11. J. S. Leon, V. Pless and N. J. A. Sloane, Self-dual codes over GF(5), J. Combin. Theory Ser. A 32 (1982), 178-194. https://doi.org/10.1016/0097-3165(82)90019-X
12. F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, North-Holland, Amsterdam, 1977.
13. C. L. Mallows, V. Pless and N. J. A. Sloane, Self-dual codes over GF(3), Siam J. Appl. Math. 31 (1976), 649-666. https://doi.org/10.1137/0131058
14. K. Nagata, F. Nemenzo and H. Wada, On self-dual codes over $Z_{16}$, Lecture Notes in Computer Science 5527 (2009), 107-116.
15. K. Nagata, F. Nemenzo and H.Wada, Constructive algorithm of self-dual error- correcting codes, Eleventh international workshop on algebraic and combinato- rial coding theory, June 16-22, Bulgaria, 215-220, 2008.
16. G. Nebe, E. Rains and N. J. A. Sloane, Self-dual codes and invariant theory, Springer-Verlag, 2006.
17. Y. H. Park, Modular independence and generator matrices for codes over $Z_{m}$, Des. Codes. Crypt 50 (2009), 147-162. https://doi.org/10.1007/s10623-008-9220-8
18. Y. H. Park, The classification of self-dual modular codes, submitted, 2010
19. V. Pless and V. D. Tonchev, Self-dual codes over GF(7), IEEE Trans. Inform. Theory 33 (1987), 723-727. https://doi.org/10.1109/TIT.1987.1057345
20. E. Rains and N. J. A. Sloane, Self-dual codes, in the Handbook of Coding Theory, V.S. Pless and W.C. Huffman, eds., Elsevier, Amsterdam, 1998, 177- 294.