A new method for constructing linear codes with small hulls *

Liqin Qian (1) , Xiwang Cao (1) , Wei Lu (1) , Patrick Solé (2)

1 NUAA - Nanjing University of Aeronautics and Astronautics [Nanjing]
2 I2M - Institut de Mathématiques de Marseille

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasiabelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl.

Mots clés: Linear codes; hull of a code; LCD codes; Gauss sums; quasi-abelian codes; double circulant codes MSC(2010): 94B05

Domaines: Théorie de l'information [cs.IT]; Théorie de l'information et codage [math.IT]

Liqin Qian, Xiwang Cao, Wei Lu, Patrick Solé. A new method for constructing linear codes with small hulls *. Designs, Codes and Cryptography, 2022. ⟨hal-03833912⟩