Codes over non-unitary rings have been studied recently. In particular, codes over the commutative non-unitary ring \(I_p\) (in the classification of Fine) of order \(p^2\) where p is a prime are being considered. For \(p=2\) (resp. \(p=3\) ), three categories of codes over \(I_p\) have been studied: self-orthogonal codes, quasi self-dual codes, and self-dual codes over \(I_p\) . Using some related mass formulas and building-up constructions, classifications of these codes have been done up to the permutation equivalence (resp. the monomial equivalence) for certain small lengths. In this paper, we take the prime \(p=5\) and consider the ring \(I_5\) . We introduce the notion of linear codes over \(I_5\) . We also define the same three categories of linear \(I_5\) -codes, study the structures of these \(I_5\) -codes and relate them to their associated residue and torsion codes. We classify the three categories of codes completely in lengths at most 4 up to the monomial equivalence for a given type \(\{k_1, k_2\}\) . Moreover, in the paper of Alahmadi et al. regarding the mass formula for self-orthogonal codes over \(I_p\) , mistakes in the classification of quasi self-dual codes over \(I_5\) had been made such as incorrect automorphism group order of some codes or inconsistency with the mass formula for self-orthogonal codes over \(I_p\) for length \(n=2\) and type \(\{1, 0\}\) and for length \(n=3\) and type \(\{1, 1\}\) . We correct and improve such results.
