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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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