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; namelyI, E, andHas denoted in (Fine, 1993). A new notion of duality is introduced in the case of the non-commutative ringE. The notion of self-dual codes with respect to this duality coincides with that of quasi self-dual codes overEas introduced in (Alahmadiet al, 2022). We characterize self-dual codes and LCD codes over the three rings, and investigate the properties of their corresponding additive codes over F4. 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 overE.