Optimal Binary Few-Weight Codes Using a Mixed Alphabet Ring and Simplicial Complexes

We construct several families of distance-optimal few-weight binary linear codes. As a method, we use the mixed alphabet ring Z 2 Z 2 [ u ], u 2 = 0 (viewing Z 2 Z 2 [ u ] as a Z 2 [ u ]-module) and three suitable defining sets, each consisting of three simplicial complexes generated by a single maximal element to construct three different families of linear codes over Z 2 [ u ], u 2 = 0. We explicitly determine their Lee weight distributions and study their Gray images to obtain our results. It turns out that most of the distance-optimal codes obtained in this paper are self-orthogonal and minimal as well. We emphasize that we find an infinite family of binary three-weight projective codes with new parameters, which produce strongly l -walk-regular graphs for every odd l ≥ 3.