Classification of LCD and self-dual codes over a finite non-unital local ring
Anup Kushwaha, Indibar Debnath, Om Prakash
TL;DR
This work studies LCD and self-dual codes over the nonunital, noncommutative ring $E_p$ of order $p^2$, linking codes over $E_p$ to their residue codes over $\mathbb{F}_p$ via a generator form $rG$ and residue/torsion structures. It establishes that all $E_p$-LCD codes are free, reduces monomial equivalence to residue-code equivalence over $\mathbb{F}_p$, and derives conditions for $E_p$-codes to be MDS/AMDS; it then classifies and enumerates LCD codes over $E_2$ and $E_3$ for short lengths, and provides initial classifications of left self-dual and self-dual codes with corresponding MDS/AMDS properties. The results extend finite-field LCD/self-dual theory to a nonunital ring setting, offering explicit constructions and tables (via MAGMA) that connect $E_p$-codes to their $\mathbb{F}_p$ counterparts. This advances coding theory over nonunital rings and suggests practical implications for secure data storage and transmission in ring-based alphabets.
Abstract
This work explores LCD and self-dual codes over a noncommutative non-unital ring $ E_p= \langle r,s ~|~ pr =ps=0,~ r^2=r,~ s^2=s,~ rs=r,~ sr=s \rangle$ of order $p^2$ where $p$ is a prime. Initially, we study the monomial equivalence of two free $E_p$-linear codes. In addition, a necessary and sufficient condition is derived for a free $E_p$-linear code to be MDS and almost MDS (AMDS). Then, we use these results to classify MDS and AMDS LCD codes over $E_2$ and $E_3$ under monomial equivalence for lengths up to $6$. Subsequently, we study left self-dual codes over the ring $E_p$ and classify MDS and AMDS left self-dual codes over $E_2$ and $E_3$ for lengths up to $12$. Finally, we study self-dual codes over the ring $E_p$ and classify MDS and AMDS self-dual codes over $E_2$ and $E_3$ for smaller lengths.