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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

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.  Google Scholar

[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.  Google Scholar

[3]

M. Grassl, Bounds on the minimum distance of linear codes,, available online at , ().   Google Scholar

[4]

W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511807077.  Google Scholar

[5]

T. Kasami, Optimum shortened cyclic codes for burst-error correction, IEEE Trans. Inform. Theory, 9 (1963), 105-109.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. Google Scholar

[9]

M. Matsuoka, $\theta$-polycyclic codes and $\theta$-sequential codes over finite fields, Int. J. Algebra, 5 (2011), 65-70.  Google Scholar

[10]

W. W. Peterson and E. J. Weldon, Error Correcting Codes, MIT Press, 1972.  Google Scholar

[11]

E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, Elsevier, Amsterdam, 1998.  Google Scholar

[12]

J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

M. Grassl, Bounds on the minimum distance of linear codes,, available online at , ().   Google Scholar

[4]

W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511807077.  Google Scholar

[5]

T. Kasami, Optimum shortened cyclic codes for burst-error correction, IEEE Trans. Inform. Theory, 9 (1963), 105-109.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. Google Scholar

[9]

M. Matsuoka, $\theta$-polycyclic codes and $\theta$-sequential codes over finite fields, Int. J. Algebra, 5 (2011), 65-70.  Google Scholar

[10]

W. W. Peterson and E. J. Weldon, Error Correcting Codes, MIT Press, 1972.  Google Scholar

[11]

E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, Elsevier, Amsterdam, 1998.  Google Scholar

[12]

J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.  Google Scholar

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