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

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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

Amita Sahni ^1, and Poonam Trama Sehgal ^1,

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

Received October 2013 Revised December 2014 Published November 2015

Abstract
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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.

Keywords: self-dual, negacyclic codes, Generator polynomial, self-orthogonal..

Mathematics Subject Classification: 11T71, 94B0.

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

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.
[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.
[3]	T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943. doi: 10.1016/j.ffa.2008.05.004.
[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.
[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.
[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.
[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.
[8]	W. Cary Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077.
[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.
[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.
[11]	F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.
[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.

show all references

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.
[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.
[3]	T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943. doi: 10.1016/j.ffa.2008.05.004.
[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.
[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.
[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.
[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.
[8]	W. Cary Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077.
[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.
[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.
[11]	F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.
[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.

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