Existence conditions for self-orthogonal negacyclic codes over finite fields

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February 2015, 9(1): 1-7. doi: 10.3934/amc.2015.9.1

Existence conditions for self-orthogonal negacyclic codes over finite fields

Liren Lin ^1, , Hongwei Liu ^2, and Bocong Chen ^3,

1.	Department of Physical Science and Technology, Central China Normal University, Wuhan, Hubei 430079, China
2.	School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China
3.	School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore 637616, Singapore

Received May 2013 Revised April 2014 Published February 2015

Abstract
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In this paper, we obtain necessary and sufficient conditions for the nonexistence of nonzero self-orthogonal negacyclic codes over a finite field, of length relatively prime to the characteristic of the underlying field.

Keywords: cyclotomic coset., self-orthogonal code, negacyclic code, Cyclic code.

Mathematics Subject Classification: Primary: 11T71; Secondary: 94B1.

Citation: Liren Lin, Hongwei Liu, Bocong Chen. Existence conditions for self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (1) : 1-7. doi: 10.3934/amc.2015.9.1

References:

[1]	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.
[2]	T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943. doi: 10.1016/j.ffa.2008.05.004.
[3]	I. F. Blake, S. Gao and R. C. Mullin, Explicit factorization of $X^{2^k}+1$ over $F_p$ with prime $p\equiv3 (mod 4)$, Appl. Algebra Engrg. Comm. Comput., 4 (1993), 89-94. doi: 10.1007/BF01386832.
[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 Math., 313 (2013), 983-991. doi: 10.1016/j.disc.2013.01.024.
[6]	W. Fu and T. Feng, On self-orthogonal group ring codes, Designs Codes Crypt., 50 (2009), 203-214. doi: 10.1007/s10623-008-9224-4.
[7]	W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl., 11 (2005), 451-490. doi: 10.1016/j.ffa.2005.05.012.
[8]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
[9]	Y. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory, 57 (2011), 2243-2251. doi: 10.1109/TIT.2010.2092415.
[10]	X. Kai and S. Zhu, On cyclic self-dual codes, Appl. Algebra Engrg. Comm. Comput., 19 (2008), 509-525. doi: 10.1007/s00200-008-0086-9.
[11]	L. Kathuria and M. Raka, Existence of cyclic self-orthogonal codes: a note on a result of Vera Pless, Adv. Math. Commun., 6 (2012), 499-503. doi: 10.3934/amc.2012.6.499.
[12]	R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 2008.
[13]	V. Pless, Cyclotomy and cyclic codes, the unreasonable effectiveness of number theory, in Proc. Sympos. Appl. Math., Amer. Math. Soc., 1992, 91-104.
[14]	N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes, IEEE Trans. Inf. Theory, 29 (1983), 364-367. doi: 10.1109/TIT.1983.1056682.
[15]	Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, 2003.
[16]	W. Willems, A note on self-dual group codes, IEEE Trans. Inf. Theory, 48 (2002), 3107-3109. doi: 10.1109/TIT.2002.805076.

show all references

References:

[1]	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.
[2]	T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943. doi: 10.1016/j.ffa.2008.05.004.
[3]	I. F. Blake, S. Gao and R. C. Mullin, Explicit factorization of $X^{2^k}+1$ over $F_p$ with prime $p\equiv3 (mod 4)$, Appl. Algebra Engrg. Comm. Comput., 4 (1993), 89-94. doi: 10.1007/BF01386832.
[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 Math., 313 (2013), 983-991. doi: 10.1016/j.disc.2013.01.024.
[6]	W. Fu and T. Feng, On self-orthogonal group ring codes, Designs Codes Crypt., 50 (2009), 203-214. doi: 10.1007/s10623-008-9224-4.
[7]	W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl., 11 (2005), 451-490. doi: 10.1016/j.ffa.2005.05.012.
[8]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
[9]	Y. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory, 57 (2011), 2243-2251. doi: 10.1109/TIT.2010.2092415.
[10]	X. Kai and S. Zhu, On cyclic self-dual codes, Appl. Algebra Engrg. Comm. Comput., 19 (2008), 509-525. doi: 10.1007/s00200-008-0086-9.
[11]	L. Kathuria and M. Raka, Existence of cyclic self-orthogonal codes: a note on a result of Vera Pless, Adv. Math. Commun., 6 (2012), 499-503. doi: 10.3934/amc.2012.6.499.
[12]	R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 2008.
[13]	V. Pless, Cyclotomy and cyclic codes, the unreasonable effectiveness of number theory, in Proc. Sympos. Appl. Math., Amer. Math. Soc., 1992, 91-104.
[14]	N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes, IEEE Trans. Inf. Theory, 29 (1983), 364-367. doi: 10.1109/TIT.1983.1056682.
[15]	Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, 2003.
[16]	W. Willems, A note on self-dual group codes, IEEE Trans. Inf. Theory, 48 (2002), 3107-3109. doi: 10.1109/TIT.2002.805076.

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