# CONSTRUCTION OF SELF-DUAL CODES OVER F2 + uF2

• Han, Sung-Hyu (School of Liberal Arts Korea University of Technology and Education) ;
• Lee, Hei-Sook (Department of Mathematics Ewha Womans University) ;
• Lee, Yoon-Jin (Department of Mathematics Ewha Womans University)
• Published : 2012.01.31

#### Abstract

We present two kinds of construction methods for self-dual codes over $\mathbb{F}_2+u\mathbb{F}_2$. Specially, the second construction (respectively, the first one) preserves the types of codes, that is, the constructed codes from Type II (respectively, Type IV) is also Type II (respectively, Type IV). Every Type II (respectively, Type IV) code over $\mathbb{F}_2+u\mathbb{F}_2$ of free rank larger than three (respectively, one) can be obtained via the second construction (respectively, the first one). Using these constructions, we update the information on self-dual codes over $\mathbb{F}_2+u\mathbb{F}_2$ of length 9 and 10, in terms of the highest minimum (Hamming, Lee, or Euclidean) weight and the number of inequivalent codes with the highest minimum weight.

#### References

1. C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Combin. Theory Ser. A 78 (1997), no. 1, 92-119. https://doi.org/10.1006/jcta.1996.2763
2. J. Cannon and C. Playoust, An Introduction to Magma, University of Sydney, Sydney, Australia, 1994.
3. S. T. Dougherty, P. Gaborit, M. Harada, and P. Sole, Type II codes over ${\mathbb{F}}_{2}+u{{\mathbb{F}}_{2}$, IEEE Trans. Inform. Theory 45 (1999), no. 1, 32-45. https://doi.org/10.1109/18.746770
4. S. T. Dougherty, P. Gaborit, M. Harada, A. Munemasa, and P. Sole, Type IV self-dual codes over rings, IEEE Trans. Inform. Theory 45 (1999), no. 7, 2345-2360. https://doi.org/10.1109/18.796375
5. A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, The ${\mathbb{Z}}_{4}$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40 (1994), no. 2, 301-319. https://doi.org/10.1109/18.312154
6. S. Han, online available at http://kutacc.kut.ac.kr/-sunghyu/data/sdF2uF2.htm
7. W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl. 11 (2005), no. 3, 451-490. https://doi.org/10.1016/j.ffa.2005.05.012
8. J.-L. Kim and Y. Lee, Euclidean and Hermitian self-dual MDS codes over large finite fields, J. Combin. Theory Ser. A 105 (2004), no. 1, 79-95. https://doi.org/10.1016/j.jcta.2003.10.003
9. J.-L. Kim and Y. Lee, Building-up constructions for self-dual codes, preprint.
10. H. Lee and Y. Lee, Construction of self-dual codes over finite rings $Z_{p}m$, J. Combin. Theory Ser. A 115 (2008), no. 3, 407-422. https://doi.org/10.1016/j.jcta.2007.07.001

#### Cited by

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2. Hermitian self-dual codes over vol.25, 2014, https://doi.org/10.1016/j.ffa.2013.08.004
3. AN EFFICIENT CONSTRUCTION OF SELF-DUAL CODES vol.52, pp.3, 2015, https://doi.org/10.4134/BKMS.2015.52.3.915
4. On the Problem of the Existence of a Square Matrix U Such That UUT=-I over Zpm vol.8, pp.3, 2017, https://doi.org/10.3390/info8030080