Average dimensions of Galois hulls of constacyclic codes

The dimensions of hulls of linear codes have a critical role in determining the complexities of the algorithms used to find solutions to some of the important coding theory problems, such as checking the permutation equivalence of two linear codes. In this paper, we provide the formula to calculate the average dimensions of Galois hulls of constacyclic codes over finite fields. Further, we define the Galois inner product and study the Galois duals of linear codes over the ring $ R_{m, q} = \frac{\mathbb{F}_q[u]}{\langle u^m-u\rangle} $. Then we derive formulae for each, the Galois hull dimensions and the average Galois hull dimensions of constacyclic codes over $ R_{m, q} $. Moreover, we provide a construction for the Galois LCD constacyclic codes over $ R_{m, q} $ and include some supportive examples of obtained codes.