Linear codes with small hulls in semi-primitive case

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The hull plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. It has been shown that these algorithms are very effective in general if the size of the hull is small. It is clear that the linear codes with the smallest hull are LCD codes and with the second smallest hull are those with one-dimensional hull. In this paper, we employ character sums in semi-primitive case to construct LCD codes and linear codes with one-dimensional hull from cyclotomic fields and multiplicative subgroups of finite fields. Some sufficient and necessary conditions for these codes are obtained, where prime ideal decompositions of prime p in cyclotomic fields play a key role. In addition, we show the non-existence of these codes in some cases.