# American Institute of Mathematical Sciences

February  2020, 14(1): 35-51. doi: 10.3934/amc.2020004

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

 1 Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand 2 Algebra and Applications Research Unit, Department of Mathematics, and Statistics, Faculty of Science, Prince of Songkla University, Hatyai, Songkhla 90110, Thailand 3 Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, India

* Corresponding author

Received  May 2018 Published  August 2019

Fund Project: 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.

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.

Citation: Somphong Jitman, Supawadee Prugsapitak, Madhu Raka. Some generalizations of good integers and their applications in the study of self-dual negacyclic codes. Advances in Mathematics of Communications, 2020, 14 (1) : 35-51. doi: 10.3934/amc.2020004
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##### References:
 [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. Google Scholar [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. Google Scholar [3] T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943. doi: 10.1016/j.ffa.2008.05.004. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] Y. Jia, S. 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. Google Scholar [8] S. Jitman, Good integers and some applications in coding theory, Cryptogr. Commun., 10 (2018), 685-704. doi: 10.1007/s12095-017-0255-4. Google Scholar [9] S. Jitman, S. Ling, H. 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. Google Scholar [10] S. Jitman, S. Ling and P. Solé, Hermitian self-dual abelian codes., IEEE Trans. Inf. Theory, 60 (2014), 1496-1507. doi: 10.1109/TIT.2013.2296495. Google Scholar [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. Google Scholar [12] X. L. Li, Repeated-root self-dual negacyclic codes over finite fields, J. Math. Res. Appl., 36 (2016), 275-284. Google Scholar [13] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997. Google Scholar [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. Google Scholar [15] M. B. Nathanson, Elementary Methods in Number Theory, Springer-Verlag, New York, 2000. Google Scholar [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. Google Scholar [17] E. Sangwisut, S. Jitman, S. 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. Google Scholar [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. Google Scholar
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