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Some generalizations of good integers  and  their applications in the  study  of  self-dual negacyclic codes

  • * Corresponding author

    * Corresponding author 

S. Jitman was supported by the Thailand Research Fund and Silpakorn University under Research Grant RSA6280042. M. Raka was supported by the Council of Scientific and Industrial Research (CSIR), India, Sanction No. 21(1042)/17/EMR-II.

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  • Good integers introduced in 1997 form an interesting family of integers that has been continuously studied  due to their rich number theoretical   properties and wide applications. In this paper, we have focused on classes  of  $ 2^\beta $-good integers, $ 2^\beta $-oddly-good integers, and $ 2^\beta $-evenly-good integers which are  generalizations  of good integers.  Properties of such integers have been given as well as their applications in   characterizing and enumerating self-dual negacyclic codes over finite fields.  An alternative proof for  the characterization of  the existence of a  self-dual negacyclic code over finite fields has been given in terms of such generalized good integers.  A general enumeration formula for the number of   self-dual negacyclic codes of length $ n $  over  finite fields  has been established.  For some specific lengths, explicit formulas have been provided as well.  Some known results on self-dual negacyclic codes over finite fields can be formalized and viewed as special cases of this work.

    Mathematics Subject Classification: Primary: 11N25, 94B15; Secondary: 94B60.

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  • [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., 19 (2013), 39-54.  doi: 10.1016/j.ffa.2012.10.003.
    [3] T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943.  doi: 10.1016/j.ffa.2008.05.004.
    [4] H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb{F}_{p^m} + \mathbb{F}_{p^m} $, J. Algebra, 324 (2010), 940-950.  doi: 10.1016/j.jalgebra.2010.05.027.
    [5] 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.
    [6] K. Guenda and T. A. Gulliver, Self-dual repeated root cyclic and negacyclic codes over finite fields, 2012 IEEE International Symposium on Information Theory Proceedings, Cambridge, MA, (2012), 2904–2908. doi: 10.1109/ISIT.2012.6284057.
    [7] Y. JiaS. Ling and C. P. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory, 57 (2011), 2243-2251.  doi: 10.1109/TIT.2010.2092415.
    [8] S. Jitman, Good integers and some applications in coding theory, Cryptogr. Commun., 10 (2018), 685-704.  doi: 10.1007/s12095-017-0255-4.
    [9] S. JitmanS. LingH. W. Liu and X. L. Xie, Abelian codes in principal ideal group algebras, IEEE Trans. Inform. Theory, 59 (2013), 3046-3058.  doi: 10.1109/TIT.2012.2236383.
    [10] S. JitmanS. Ling and P. Solé, Hermitian self-dual abelian codes., IEEE Trans. Inf. Theory, 60 (2014), 1496-1507.  doi: 10.1109/TIT.2013.2296495.
    [11] D. Knee and H. D. Goldman, Quasi-self-reciprocal polynomials and potentially large minimum distance BCH codes, IEEE Trans. Inform. Theory, 15 (1969), 118-121.  doi: 10.1109/tit.1969.1054262.
    [12] X. L. Li, Repeated-root self-dual negacyclic codes over finite fields, J. Math. Res. Appl., 36 (2016), 275-284. 
    [13] R. Lidl and  H. NiederreiterFinite Fields, Cambridge University Press, Cambridge, 1997. 
    [14] P. Moree, On the divisors of $a^k+b^k$, Acta Arithmetica, 80 (1997), 197-212.  doi: 10.4064/aa-80-3-197-212.
    [15] M. B. Nathanson, Elementary Methods in Number Theory, Springer-Verlag, New York, 2000.
    [16] M. Raka, A class of constacyclic codes over a finite field-II, Indian J. Pure Appl. Math., 46 (2015), 809-825.  doi: 10.1007/s13226-015-0158-z.
    [17] E. SangwisutS. JitmanS. Ling and P. Udomkavanich, Hulls of cyclic and negacyclic codes over finite fields, Finite Fields Appl., 33 (2015), 232-257.  doi: 10.1016/j.ffa.2014.12.008.
    [18] Y. S. Yang and W. C. Cai, On self-dual constacyclic codes over finite fields, Des. Codes Cryptogr., 74 (2015), 355-364.  doi: 10.1007/s10623-013-9865-9.
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