On the duality and the direction of polycyclic codes

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November 2016, 10(4): 921-929. doi: 10.3934/amc.2016049

On the duality and the direction of polycyclic codes

Adel Alahmadi ^1, , Steven Dougherty ^2, , André Leroy ^3, and Patrick Solé ^4,

1.	Math Dept., King Abdulaziz University, Jeddah, Saudi Arabia
2.	University of Scranton, Scranton, PA 18518, United States
3.	University of Artois, Faculté J. Perrin, 62300 Lens, France
4.	CNRS/LAGA, University of Paris 8, 93 526 Saint-Denis, France

Received March 2015 Published November 2016

Abstract
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Polycyclic codes are ideals in quotients of polynomial rings by a principal ideal. Special cases are cyclic and constacyclic codes. A MacWilliams relation between such a code and its annihilator ideal is derived. An infinite family of binary self-dual codes that are also formally self-dual in the classical sense is exhibited. We show that right polycyclic codes are left polycyclic codes with different (explicit) associate vectors and characterize the case when a code is both left and right polycyclic for the same associate polynomial. A similar study is led for sequential codes.

Keywords: Cyclic codes, formally self-dual codes..

Mathematics Subject Classification: Primary: 94B15, 94B05, 94B06.

Citation: Adel Alahmadi, Steven Dougherty, André Leroy, Patrick Solé. On the duality and the direction of polycyclic codes. Advances in Mathematics of Communications, 2016, 10 (4) : 921-929. doi: 10.3934/amc.2016049

References:

[1]	K. Betsumiya and M. Harada, Binary optimal odd formally self-dual codes, Des. Codes Crypt., 23 (2001), 11-22. doi: 10.1023/A:1011203416769.
[2]	J. Fields, P. Gaborit, V. Pless and W. C. Huffman, On the classification of extremal even formally self-dual codes of lengths $20$ and $22$, Discrete Appl. Math., 111 (2001), 75-86. doi: 10.1016/S0166-218X(00)00345-0.
[3]	M. Grassl, Bounds on the minimum distance of linear codes,, available online at , ().
[4]	W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511807077.
[5]	T. Kasami, Optimum shortened cyclic codes for burst-error correction, IEEE Trans. Inform. Theory, 9 (1963), 105-109.
[6]	J.-L. Kim and V. Pless, A note on formally self-dual even codes of length divisible by 8, Finite Fields Appl., 13 (2007), 224-229. doi: 10.1016/j.ffa.2005.09.006.
[7]	S. R. Lopez-Permouth, B. R. Parra-Avila and S. Szabo, Dual generalizations of the concept of cyclicity of codes, Adv. Math. Commun., 3 (2009), 227-234. doi: 10.3934/amc.2009.3.227.
[8]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.
[9]	M. Matsuoka, $\theta$-polycyclic codes and $\theta$-sequential codes over finite fields, Int. J. Algebra, 5 (2011), 65-70.
[10]	W. W. Peterson and E. J. Weldon, Error Correcting Codes, MIT Press, 1972.
[11]	E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, Elsevier, Amsterdam, 1998.
[12]	J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.

show all references

References:

[1]	K. Betsumiya and M. Harada, Binary optimal odd formally self-dual codes, Des. Codes Crypt., 23 (2001), 11-22. doi: 10.1023/A:1011203416769.
[2]	J. Fields, P. Gaborit, V. Pless and W. C. Huffman, On the classification of extremal even formally self-dual codes of lengths $20$ and $22$, Discrete Appl. Math., 111 (2001), 75-86. doi: 10.1016/S0166-218X(00)00345-0.
[3]	M. Grassl, Bounds on the minimum distance of linear codes,, available online at , ().
[4]	W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511807077.
[5]	T. Kasami, Optimum shortened cyclic codes for burst-error correction, IEEE Trans. Inform. Theory, 9 (1963), 105-109.
[6]	J.-L. Kim and V. Pless, A note on formally self-dual even codes of length divisible by 8, Finite Fields Appl., 13 (2007), 224-229. doi: 10.1016/j.ffa.2005.09.006.
[7]	S. R. Lopez-Permouth, B. R. Parra-Avila and S. Szabo, Dual generalizations of the concept of cyclicity of codes, Adv. Math. Commun., 3 (2009), 227-234. doi: 10.3934/amc.2009.3.227.
[8]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.
[9]	M. Matsuoka, $\theta$-polycyclic codes and $\theta$-sequential codes over finite fields, Int. J. Algebra, 5 (2011), 65-70.
[10]	W. W. Peterson and E. J. Weldon, Error Correcting Codes, MIT Press, 1972.
[11]	E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, Elsevier, Amsterdam, 1998.
[12]	J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.

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