Linear codes with one-dimensional hull associated with Gaussian sums

Abstract

The hull of a linear code over finite fields, the intersection of the code and its dual, has been of interest and extensively studied due to its wide applications. For example, it plays a vital role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and for computing the automorphism group of a linear code. People are interested in pursuing linear codes with small hulls since, for such codes, the aforementioned algorithms are very efficient. In this field, Carlet, Mesnager, Tang and Qi gave a systematic characterization of LCD codes, i.e, linear codes with null hull. In 2019, Carlet, Li and Mesnager presented some constructions of linear codes with small hulls. In the same year, Li and Zeng derived some constructions of linear codes with one-dimensional hull by using some specific Gaussian sums. In this paper, we use general Gaussian sums to construct linear codes with one-dimensional hull by utilizing number fields, which generalizes some results of Li and Zeng (IEEE Trans. Inf. Theory 65(3), 1668–1676, 2019) and also of those presented by Carlet et al. (Des. Codes Cryptogr. 87(12), 3063–3075, 2019). We give sufficient conditions to obtain such codes. Notably, some codes we obtained are optimal or almost optimal according to the Database. This is the first attempt on constructing linear codes by general Gaussian sums which have one-dimensional hull and are optimal. Moreover, we also develop a bound of on the minimum distances of linear codes we constructed.