# American Institute of Mathematical Sciences

November  2015, 9(4): 437-447. doi: 10.3934/amc.2015.9.437

## Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields

 1 Centre for Advanced Study in Mathematics, Panjab University, Chandigarh 160014, India, India

Received  October 2013 Revised  December 2014 Published  November 2015

The main objective of this article is to study self-orthogonal negacyclic codes of length $n$ over a finite field $\mathbb{F}_{q}$, where the characteristic of $\mathbb{F}_{q}$ does not divide $n$. We investigate issues related to their existence, characterization and enumeration. We find the necessary and sufficient conditions for the existence of self-orthogonal negacyclic codes of length $n$ over a finite field $\mathbb{F}_{q}$. We characterize the defining sets and the corresponding generator polynomials of these codes. We obtain formulae to calculate the number of self-dual and self-orthogonal negacyclic codes of a given length $n$ over $\mathbb{F}_{q}$. The enumeration formula for self-orthogonal negacyclic codes involves a two-variable function $\chi(d,q)$ defined by $\chi(d,q)=0$ if $d$ divides $(q^{k}+1)$ for some $k\geq0$ and $\chi(d,q)=1$, otherwise. We give necessary and sufficient conditions when $\chi(d,q)=0$ holds.
Citation: Amita Sahni, Poonam Trama Sehgal. Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (4) : 437-447. doi: 10.3934/amc.2015.9.437
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##### References:
 [1] G. K. Bakshi and M. Raka, A class of constacyclic codes over a finite field, Finite Fields Appl., 18 (2012), 362-377. doi: 10.1016/j.ffa.2011.09.005.  Google Scholar [2] G. K. Bakshi and M. Raka, Self-dual and self-orthogonal negacyclic codes of length $2p^n$ over a finite field, Finite Fields Appl., 19 (2013), 39-54. doi: 10.1016/j.ffa.2012.10.003.  Google Scholar [3] T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943. doi: 10.1016/j.ffa.2008.05.004.  Google Scholar [4] H. Q. Dinh, Repeated-root constacyclic codes of length $2p^s$, Finite Fields Appl., 18 (2012), 133-143. doi: 10.1016/j.ffa.2011.07.003.  Google Scholar [5] H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3p^s$ and their duals, Discrete Mathematics, 313 (2013), 983-991. doi: 10.1016/j.disc.2013.01.024.  Google Scholar [6] H. Q. Dinh, On repeated-root constacyclic codes of length $4p^s$, Asian-European J. Math., 6 (2013), 1350020 [25 pages]. doi: 10.1142/S1793557113500204.  Google Scholar [7] H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744. doi: 10.1109/TIT.2004.831789.  Google Scholar [8] W. Cary Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077.  Google Scholar [9] Y. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inform. Theory, 57 (2011), 2243-2251. doi: 10.1109/TIT.2010.2092415.  Google Scholar [10] J. S. Leon and V. Pless, Self-dual codes over GF(5), Journal of Combinatorial Theory, Series A, 32 (1982), 178-194. doi: 10.1016/0097-3165(82)90019-X.  Google Scholar [11] F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. Google Scholar [12] V. Pless, A classification of self-orthogonal codes over GF(2), Discrete Mathematics, 3 (1972), 209-246. doi: 10.1016/0012-365X(72)90034-9.  Google Scholar
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