Binary LCD Codes and Their Graph Representations
For coding theorists and graph theorists, this work offers a complete theoretical characterization that unifies and extends known results, though it is incremental in nature.
This paper provides a complete characterization of simple graphs whose adjacency matrices generate binary LCD codes, including a necessary and sufficient criterion for distance-regular graphs. The results unify and strengthen previous sufficient conditions and classify all simple graphs with idempotent adjacency matrices on at most 13 vertices.
We give a complete characterization of simple graphs whose adjacency matrices generate binary linear complementary dual (LCD) codes. In particular, we completely characterize a distance-regular graph which yields an LCD code in terms of the intersection array parameters. This necessary and sufficient criterion strengthens the previously known sufficient conditions and unifies the cases of complete, Hamming, Johnson, and Grassmann graphs. As further applications, we prove that non-isomorphic conference graphs with $q \equiv 1 \pmod 8$ yield inequivalent codes and we classify all simple graphs with idempotent adjacency matrices on at most $13$ vertices via mass formulas for binary LCD codes.