Linear codes over a new commutative non-unital ring of characteristic 2 and order 16: structure, duality, and Gray map

We introduce and investigate linear codes over a new finite commutative non-unital ring

$ \mathcal{R} = \mathcal{I} + u\,\mathcal{I}, \qquad u^2 = 0, $

where

$ \mathcal{I} = \langle a,b \mid 2a = 2b = 0,\ a^2 = b,\ ab = 0 \rangle. $

The ring $ \mathcal{R} $ has characteristic $ 2 $, order $ 16 $, and a rich nilpotent structure, providing a new algebraic setting for coding theory beyond finite fields and Frobenius rings. We establish a comprehensive framework for linear codes over $ \mathcal{R} $ by describing the module structure of $ \mathcal{R} $ over $ \mathbb{F}_2 + u\mathbb{F}_2 $ and introducing residue and torsion codes. Using this residue-torsion framework, we derive characterizations of self-orthogonal, quasi-self-dual, self-dual, and free codes together with explicit descriptions of their duals. Furthermore, we construct an $ \mathbb{F}_2+u\mathbb{F}_2 $-linear Gray map and prove that it is a Lee-weight-preserving isometry from $ \mathcal{R}^n $ to $ (\mathbb{F}_2+u\mathbb{F}_2)^{2n} $. We also develop the theory of Hamming and Lee weights, establish the corresponding weight enumerators, and investigate their structural properties. Finally, several explicit examples are constructed and verified using the computational algebra system MAGMA, illustrating the existence of quasi-self-dual, self-dual, and Type Ⅳ codes together with their weight distributions. These results demonstrate that the ring $ \mathcal{R} $ provides a natural and effective framework for constructing and studying new families of linear codes over finite non-unital rings, thereby enriching the algebraic foundations of coding theory.