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
$\mathbb {Z}_{2} \mathbb {Z}_{2}[u],u^{2}=0$
(viewing
$\mathbb {Z}_{2} \mathbb {Z}_{2}[u]$
as a
$\mathbb {Z}_{2}[u]$
-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
$\mathbb {Z}_{2}[u],u^{2}=0$
. 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
$l\ge 3$
.