# 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
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##### References:
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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