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.

有限域上的线性代码的外壳，代码及其对偶的交集由于其广泛的应用而受到关注并得到了广泛的研究。例如，它在确定用于检查两个线性代码的置换等效性以及计算线性代码的自同构组的算法的复杂性方面起着至关重要的作用。人们对追求具有小船体的线性代码感兴趣，因为对于此类代码，上述算法非常有效。在该领域，Carlet，Mesnager，Tang和Qi对LCD代码（即具有空壳的线性代码）进行了系统的表征。在2019年，Carlet，Li和Mesnager提出了一些带有小船体的线性代码构造。同年，Li和Zeng通过使用一些特定的高斯和得出了带有一维船体的线性代码的一些构造。65（3），1668–1676，2019）以及Carlet等人的研究。（德。代码Cryptogr。87（12），3063-3075，2019）。我们提供了足够的条件来获取此类代码。值得注意的是，根据数据库，我们获得的某些代码是最佳的或几乎最佳的。这是通过具有一维外壳并且是最优的一般高斯和构造线性代码的首次尝试。此外，我们还对构造的线性代码的最小距离进行了限制。