American Institute of Mathematical Sciences

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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

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.  Google Scholar

[2]

T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943. doi: 10.1016/j.ffa.2008.05.004.  Google Scholar

[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.  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 Math., 313 (2013), 983-991. doi: 10.1016/j.disc.2013.01.024.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 2008. Google Scholar

[13]

V. Pless, Cyclotomy and cyclic codes, the unreasonable effectiveness of number theory, in Proc. Sympos. Appl. Math., Amer. Math. Soc., 1992, 91-104. Google Scholar

[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.  Google Scholar

[15]

Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, 2003. Google Scholar

[16]

W. Willems, A note on self-dual group codes, IEEE Trans. Inf. Theory, 48 (2002), 3107-3109. doi: 10.1109/TIT.2002.805076.  Google Scholar

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.  Google Scholar

[2]

T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943. doi: 10.1016/j.ffa.2008.05.004.  Google Scholar

[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.  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 Math., 313 (2013), 983-991. doi: 10.1016/j.disc.2013.01.024.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 2008. Google Scholar

[13]

V. Pless, Cyclotomy and cyclic codes, the unreasonable effectiveness of number theory, in Proc. Sympos. Appl. Math., Amer. Math. Soc., 1992, 91-104. Google Scholar

[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.  Google Scholar

[15]

Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, 2003. Google Scholar

[16]

W. Willems, A note on self-dual group codes, IEEE Trans. Inf. Theory, 48 (2002), 3107-3109. doi: 10.1109/TIT.2002.805076.  Google Scholar

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