Duality of Codes Over Non-Unital Rings of Order Four

In this paper, we present a basic theory of the duality of linear codes over three of the non-unital rings of order four; namely
$I$
,
$E$
, and
$H$
as denoted in (Fine, 1993). A new notion of duality is introduced in the case of the non-commutative ring
$E$
. The notion of self-dual codes with respect to this duality coincides with that of quasi self-dual codes over
$E$
as introduced in (Alahmadi et al, 2022). We characterize self-dual codes and LCD codes over the three rings, and investigate the properties of their corresponding additive codes over
$\mathop {\mathrm {\mathbb {F}}} _{4}$
. We study the connection between the dual of any linear code over these rings and the dual of its associated binary codes. A MacWilliams formula is established for linear codes over
$E$
.