@article{Alahmadi2023, 
author = {Adel Alahmadi and Asmaa Melaibari and Patrick Solé},
title = {Quasi self-dual codes over non-unital rings from three-class association schemes},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {10},
pages = {22731-22757},
keywords = {rings, self-orthogonal codes, free codes, association schemes, adjacency matrices},
url = {https://www.sciopen.com/article/10.3934/math.20231158},
doi = {10.3934/math.20231158},
abstract = {Let    E and    I denote the two non-unital rings of order 4 in the notation of (Fine, 93) defined by generators and relations as    E  =  ⟨  a  ,  b  ∣  2  a  =  2  b  =  0  ,      a    2    =  a  ,      b    2    =  b  ,  a  b  =  a  ,  b  a  =  b  ⟩ and    I  =  ⟨  a  ,  b  ∣  2  a  =  2  b  =  0  ,      a    2    =  b  ,  a  b  =  0  ⟩. Recently, Alahmadi et al classified quasi self-dual (QSD) codes over the rings    E and    I for lengths up to 12 and 6, respectively. The codes had minimum distance at most 2 in the case of    I, and 4 in the case of    E. In this paper, we present two methods for constructing linear codes over these two rings using the adjacency matrices of three-class association schemes. We show that under certain conditions the constructions yield QSD or Type Ⅳ codes. Many codes with minimum distance exceeding 4 are presented. The form of the generator matrices of the codes with these constructions prompted some new results on free codes over    E and    I.}
}