# American Institute of Mathematical Sciences

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

## On the duality and the direction of polycyclic codes

 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.
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:
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##### 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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