Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless

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November 2012, 6(4): 499-503. doi: 10.3934/amc.2012.6.499

Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless

Leetika Kathuria ^1, and Madhu Raka ^1,

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

Received January 2012 Revised June 2012 Published November 2012

Abstract
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It is of interest to know when cyclic self-orthogonal codes of length $n$ over $\mathbb F_q$ do not exist. The conditions, listed by Pless in [7] under which cyclic self-orthogonal codes can not exist, are not always sufficient. An example is given to assert this. Here we give the necessary and sufficient conditions under which cyclic self-orthogonal codes of length $n$ over $\mathbb F_q$ do not exist.

Keywords: self-orthogonal codes, Self-dual codes, order of an integer., cyclotomic cosets.

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

Citation: Leetika Kathuria, Madhu Raka. Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless. Advances in Mathematics of Communications, 2012, 6 (4) : 499-503. doi: 10.3934/amc.2012.6.499

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., (). doi: 10.1016/j.ffa.2012.10.003.
[3]	D. M. Burton, "Elementry Number Theory,'' 6^th edition, Tata McGraw-Hill, 2006.
[4]	W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge, 2003. doi: 10.1017/CBO9780511807077.
[5]	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.
[6]	X. S. Kai and S. X. Zhu, On cyclic self-dual codes, Appl. Algebra Engrg. Comm. Comput., 19 (2008), 509-525. doi: 10.1007/s00200-008-0086-9.
[7]	V. Pless, Cyclotomy and cyclic codes, the unreasonable effectiveness of number theory, in "Proc. Sympos. Appl. Math. (Orono, ME, 1991),'' Amer. Math. Soc., 46 (1992), 91-104.
[8]	E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, New York, (1998), 177-294.
[9]	N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes, IEEE Trans. Inform. Theory, 29 (1983), 364-367. doi: 10.1109/TIT.1983.1056682.

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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.
[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., (). doi: 10.1016/j.ffa.2012.10.003.
[3]	D. M. Burton, "Elementry Number Theory,'' 6^th edition, Tata McGraw-Hill, 2006.
[4]	W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge, 2003. doi: 10.1017/CBO9780511807077.
[5]	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.
[6]	X. S. Kai and S. X. Zhu, On cyclic self-dual codes, Appl. Algebra Engrg. Comm. Comput., 19 (2008), 509-525. doi: 10.1007/s00200-008-0086-9.
[7]	V. Pless, Cyclotomy and cyclic codes, the unreasonable effectiveness of number theory, in "Proc. Sympos. Appl. Math. (Orono, ME, 1991),'' Amer. Math. Soc., 46 (1992), 91-104.
[8]	E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, New York, (1998), 177-294.
[9]	N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes, IEEE Trans. Inform. Theory, 29 (1983), 364-367. doi: 10.1109/TIT.1983.1056682.

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