# American Institute of Mathematical Sciences

August  2018, 12(3): 451-463. doi: 10.3934/amc.2018027

## The average dimension of the Hermitian hull of constacyclic codes over finite fields of square order

 1 Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand 2 Department of Mathematics and Statistics, Faculty of Science, Thaksin University, Phattalung 93110, Thailand

Received  February 2017 Published  July 2018

The hulls of linear and cyclic codes have been extensively studied due to their wide applications. The dimensions and average dimension of the Euclidean hull of linear and cyclic codes have been well-studied. In this paper, the average dimension of the Hermitian hull of constacyclic codes of length $n$ over a finite field $\mathbb{F}_{q^2}$ is determined together with some upper and lower bounds. It turns out that either the average dimension of the Hermitian hull of constacyclic codes of length $n$ over $\mathbb{F}_{q^2}$ is zero or it grows the same rate as $n$. Comparison to the average dimension of the Euclidean hull of cyclic codes is discussed as well.

Citation: Somphong Jitman, Ekkasit Sangwisut. The average dimension of the Hermitian hull of constacyclic codes over finite fields of square order. Advances in Mathematics of Communications, 2018, 12 (3) : 451-463. doi: 10.3934/amc.2018027
##### References:
 [1] E. F. Assmus Jr and J. D. Key, Affine and projective planes, Discrete Math., 83 (1990), 161-187.  doi: 10.1016/0012-365X(90)90003-Z. [2] 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. [3] B. Chen, Y. Fan, L. Lin and H. Liu, Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231.  doi: 10.1016/j.ffa.2012.10.001. [4] K. Guenda, S. Jitman and T. A. Gulliver, Constructions of good entanglement-assisted quantum error correcting codes, Des. Codes Cryptogr., 86 (2018), 121-136.  doi: 10.1007/s10623-017-0330-z. [5] S. Jitman, Good integers and some applications in coding theory, Cryptogr. Commun., 10 (2018), 685-704.  doi: 10.1007/s12095-017-0255-4. [6] S. Jitman and E. Sangwisut, The average dimension of the Hermitian hull of cyclic codes over finite fields of square order, in: AIP Proceedings of ICoMEIA 2016, 1775 (2016), Article ID 030026. [7] J. Leon, Computing automorphism groups of error-correcting codes, IEEE Trans. Inform. Theory, 28 (1982), 496-511.  doi: 10.1109/TIT.1982.1056498. [8] J. Leon, Permutation group algorithms based on partition, Ⅰ: Theory and algorithms, J. Symbolic Comput., 12 (1991), 533-583.  doi: 10.1016/S0747-7171(08)80103-4. [9] 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. [10] E. Sangwisut, S. Jitman and P. Udomkavanich, Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields, Adv. Math. Commun., 11 (2017), 595-613.  doi: 10.3934/amc.2017045. [11] N. Sendrier, On the dimension of the hull, SIAM J. Appl. Math., 10 (1997), 282-293.  doi: 10.1137/S0895480195294027. [12] N. Sendrier and G. Skersys, On the computation of the automorphism group of a linear code, in: Proceedings of IEEE ISIT'2001, Washington, DC, 6 (2001), 10pp. [13] N. Sendrier, Finding the permutation between equivalent binary code, in: Proceedings of IEEE ISIT'1997, Ulm, Germany, (1997), p367. doi: 10.1109/ISIT.1997.613303. [14] N. Sendrier, Finding the permutation between equivalent codes: The support splitting algorithm, IEEE Trans. Inform. Theory, 46 (2000), 1193-1203.  doi: 10.1109/18.850662. [15] G. Skersys, The average dimension of the hull of cyclic codes, Discrete Appl. Math, 128 (2003), 275-292.  doi: 10.1016/S0166-218X(02)00451-1. [16] Y. Yang and W. 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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##### References:
 [1] E. F. Assmus Jr and J. D. Key, Affine and projective planes, Discrete Math., 83 (1990), 161-187.  doi: 10.1016/0012-365X(90)90003-Z. [2] 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. [3] B. Chen, Y. Fan, L. Lin and H. Liu, Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231.  doi: 10.1016/j.ffa.2012.10.001. [4] K. Guenda, S. Jitman and T. A. Gulliver, Constructions of good entanglement-assisted quantum error correcting codes, Des. Codes Cryptogr., 86 (2018), 121-136.  doi: 10.1007/s10623-017-0330-z. [5] S. Jitman, Good integers and some applications in coding theory, Cryptogr. Commun., 10 (2018), 685-704.  doi: 10.1007/s12095-017-0255-4. [6] S. Jitman and E. Sangwisut, The average dimension of the Hermitian hull of cyclic codes over finite fields of square order, in: AIP Proceedings of ICoMEIA 2016, 1775 (2016), Article ID 030026. [7] J. Leon, Computing automorphism groups of error-correcting codes, IEEE Trans. Inform. Theory, 28 (1982), 496-511.  doi: 10.1109/TIT.1982.1056498. [8] J. Leon, Permutation group algorithms based on partition, Ⅰ: Theory and algorithms, J. Symbolic Comput., 12 (1991), 533-583.  doi: 10.1016/S0747-7171(08)80103-4. [9] 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. [10] E. Sangwisut, S. Jitman and P. Udomkavanich, Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields, Adv. Math. Commun., 11 (2017), 595-613.  doi: 10.3934/amc.2017045. [11] N. Sendrier, On the dimension of the hull, SIAM J. Appl. Math., 10 (1997), 282-293.  doi: 10.1137/S0895480195294027. [12] N. Sendrier and G. Skersys, On the computation of the automorphism group of a linear code, in: Proceedings of IEEE ISIT'2001, Washington, DC, 6 (2001), 10pp. [13] N. Sendrier, Finding the permutation between equivalent binary code, in: Proceedings of IEEE ISIT'1997, Ulm, Germany, (1997), p367. doi: 10.1109/ISIT.1997.613303. [14] N. Sendrier, Finding the permutation between equivalent codes: The support splitting algorithm, IEEE Trans. Inform. Theory, 46 (2000), 1193-1203.  doi: 10.1109/18.850662. [15] G. Skersys, The average dimension of the hull of cyclic codes, Discrete Appl. Math, 128 (2003), 275-292.  doi: 10.1016/S0166-218X(02)00451-1. [16] Y. Yang and W. Cai, On self-dual constacyclic codes over finite fields, Des. Codes Cryptogr., 74 (2015), 355-364.  doi: 10.1007/s10623-013-9865-9.
The lower and upper bounds for $E_H(n = \overline{n}p^\nu, \lambda, q^2)$
 Order of $\lambda$ Conditions LB UB Remarks $r$ is odd and $r|(q+1).$ $r\in M_q$ and $n\in M_q$ $0$ $0$ Remark 2 $r\in M_q$ and $n\not\in M_q$ $\cfrac{n}{8}$ $\cfrac{n}{3}$ Theorem 5.1 $r\not\in M_q$ $\cfrac{n}{4}$ $\cfrac{n}{3}$ $r$ is even and $r|(q+1).$ $\beta \left( {\bar{n}} \right)+\beta \left( r \right)\leq\gamma$, $r\in M_q$ and $n\in M_q$ $0$ $0$ Theorem 5.1 $\beta \left( {\bar{n}} \right)+\beta \left( r \right)\leq\gamma$, $r\in M_q$ and $n\not\in M_q$ $\dfrac{n}{6}$ $\dfrac{n}{3}$ $\beta \left( {\bar{n}} \right)+\beta \left( r \right)>\gamma$ or $r\not\in M_q$ $\dfrac{n}{4}$ $\dfrac{n}{3}$
 Order of $\lambda$ Conditions LB UB Remarks $r$ is odd and $r|(q+1).$ $r\in M_q$ and $n\in M_q$ $0$ $0$ Remark 2 $r\in M_q$ and $n\not\in M_q$ $\cfrac{n}{8}$ $\cfrac{n}{3}$ Theorem 5.1 $r\not\in M_q$ $\cfrac{n}{4}$ $\cfrac{n}{3}$ $r$ is even and $r|(q+1).$ $\beta \left( {\bar{n}} \right)+\beta \left( r \right)\leq\gamma$, $r\in M_q$ and $n\in M_q$ $0$ $0$ Theorem 5.1 $\beta \left( {\bar{n}} \right)+\beta \left( r \right)\leq\gamma$, $r\in M_q$ and $n\not\in M_q$ $\dfrac{n}{6}$ $\dfrac{n}{3}$ $\beta \left( {\bar{n}} \right)+\beta \left( r \right)>\gamma$ or $r\not\in M_q$ $\dfrac{n}{4}$ $\dfrac{n}{3}$
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