Linear codes with one-dimensional hull associated with Gaussian sums
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Linear codes with one-dimensional hull associated with Gaussian sums Liqin Qian1 · Xiwang Cao1,2 · Sihem Mesnager3,4,5 Received: 8 April 2020 / Accepted: 4 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Linear code · Hull · Gaussian sum · Number field Mathematics Subject Classification (2010) 94B05 · 11T24 · 11T71
This research is supported by the National Natural Science Foundation of China under Grant 11771007 and Grant 61572027. Xiwang Cao
[email protected]
Extended author information available on the last page of the article.
Cryptography and Communications
1 Introduction Throughout this paper, Fq denotes the finite field of order q, where q is a power of prime p. F∗q the group of the multiplicaLet F∗q = Fq \{0} denote the multiplicative group of Fq and tive characters of F∗q . Let n be a positive integer and Fnq an n dimensional vector space over Fq . An [n, k, d]q linear code C over Fq is a k-dimensional subspace of Fnq with minimum distance d. For any two vectors x = (x1 , x2 , · · · , xn ) and y = (y1 , y2 , · · · , yn ) of Fnq , their Euclidean inner product is defined as x·y =
n
xi yi .
i=1
Let C be a linear code of length n over Fq . The Euclidean dual code of C, denoted by C ⊥ , is defined by the set C ⊥ = {b ∈ Fnq : b · c = 0 for all c ∈ C}. The hull of C is the code C∩C ⊥ , denoted by Hull(C), was introduced in 1990 by Assmus and Key [1] to classify finite projective planes. Suppose that the dimension of Hull(C) is . If = 0,
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