LCD and ACD codes over a non commutative non-unital ring with four elements

Minjia Shi (1) , Shitao Li (1) , Jon-Lark Kim (2) , Patrick Solé (3)

1 Anhui University [Hefei]
2 Sogang University [Séoul]
3 I2M - Institut de Mathématiques de Marseille

We study LCD (linear complementary dual) and ACD (additive complementary dual) codes over a noncommutative non-unital ring E with four elements. This is the rst attempt to construct LCD codes over a noncommutative non-unital ring. We show that free LCD codes over E are directly related to binary LCD codes. We introduce ACD codes over E. They include free LCD codes over E as a special case. These facts imply that LCD and ACD codes over E are worth studying. In particular, we characterize a free LCD E-code C in terms of a binary generator matrix G. We also dene an ACD code over E, called a left-ACD code. We give several conditions for the existence of left-ACD codes.

Mots clés: codes; LCD codes; non-unitary ring

Domaines: Théorie de l'information [cs.IT]

Minjia Shi, Shitao Li, Jon-Lark Kim, Patrick Solé. LCD and ACD codes over a non commutative non-unital ring with four elements. Cryptography and Communications - Discrete Structures, Boolean Functions and Sequences , 2022. ⟨hal-03390420⟩