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

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 first 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 define an ACD code over E, called a left-ACD code. We give several conditions for the existence of left-ACD codes.

Automated summary

This study is the first attempt to construct LCD codes over a noncommutative non-unital ring and introduces ACD codes over the same ring. The research shows a direct relationship between free LCD codes over E and binary LCD codes, while also demonstrating that ACD codes include free LCD codes as a special case. These findings indicate that studying LCD and ACD codes over E is valuable, with specific characterization of free LCD E-codes and conditions for the existence of left-ACD codes.