Classification of LCD and self-dual codes over a finite non-unital local ring

Abstract

This work explores LCD and self-dual codes over a noncommutative non-unital ring [Formula: see text] of order [Formula: see text] where [Formula: see text] is a prime. Initially, we study the monomial equivalence of two free [Formula: see text]-linear codes. In addition, a necessary and sufficient condition is derived for a free [Formula: see text]-linear code to be maximum distance separable (MDS) and almost MDS ( AMDS). Then, we use these results to classify MDS and AMDS LCD codes over [Formula: see text] and [Formula: see text] under monomial equivalence for lengths up to [Formula: see text]. Subsequently, we study left self-dual codes over the ring [Formula: see text] and classify MDS and AMDS left self-dual codes over [Formula: see text] and [Formula: see text] for lengths up to [Formula: see text]. Finally, we study self-dual codes over the ring [Formula: see text] and classify MDS and AMDS self-dual codes over [Formula: see text] and [Formula: see text] for short lengths.