New extremal binary self-dual codes of length 68

Abstract

In this correspondence, we consider quadratic double and bordered double circulant construction methods over the ring R := F_2 + uF_2 + u^2F_2, where u^3 = 1. Among other examples, extremal binary self-dual codes of length 66 are obtained by these constructions. These are extended by using extension theorems for self-dual codes and as a result 8 new extremal binary self-dual codes of length 68 are obtained. More precisely, codes with beta=117, 120, 133 in W68;1 and with gamma = 1, beta=49, 57, 59 and codes with gamma=2, beta=69, 81 in W68;2 are constructed for the ?first time in the literature. In addition to these, some known such codes are reconstructed via this extension. The results are tabulated.

Keywords

• Extremal codes, Codes over rings, Gray maps, Quadratic double-circulant codes

References

1. J. H. Conway, N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36(6), 1319-1333, 1990.
2. S. T. Dougherty, T. A. Gulliver, M., Harada, Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43(6), 2036-2047, 1997.
3. P. Gaborit, Quadratic double circulant codes over fields, Journal of Combinatorial Theory Series A, 97(1), 85-107, 2002.
4. W.C. Huffman, V. Pless, Fundamentals of error correcting codes, Cambridge University press, 2003.
5. J.-L. Kim, New extremal self-dual codes of lengths 36, 38 and 58, IEEE Trans. Inf. Theory, 47(1), 386-393, 2001.
6. A. Kaya, B. Yildiz, İ. Şiap, New extremal binary self-dual codes from F4+ uF4-lifts of quadratic double circulant codes over F, available online at http://arxiv.org/abs/1405.7147
7. A. Kaya, B. Yildiz, Binary generator matrices of new extremal self-dual binary codes of length 68, available online at http://www.fatih.edu.tr/~akaya/binary/68u31.txt
8. A. Kaya, B. Yildiz, Extension theorems for self-dual codes over rings and new binary self-dual codes, available online at http://arxiv.org/abs/1404.0195.

9. E. M. Rains, Shadow Bounds for Self Dual Codes, IEEE Trans. Inf. Theory, 44(1), 134-139, 1998.
10. H.-P. Tsai, P.-Y. Shih, R.-Y. Wuh, W.-K. Su, C.-H. Chen, Construction of self-dual codes, IEEE
11. Trans. Inform. Theory, 54(8), 3826-3831, 2008.