  In this article, we construct linear codes over the commutative non-unital
ring $I$ of size four. We obtain their Lee-weight distributions and study their
binary Gray images. Under certain mild conditions, these classes of binary
codes are minimal and self-orthogonal. All codes in this article are few-weight
codes. Besides, an infinite class of these binary codes consists of distance
optimal codes with respect to the Griesmer bound.
