Construct an infinite Singleton-optimal triple-node code family

Construct an infinite family of binary graph codes on complete looped graphs that correct every three-node erasure and attain the graph Singleton redundancy of 3n-3, rather than the 3n-2 redundancy achieved by the cyclic triple-node constructions.

Background

The paper constructs binary triple-node-erasure-correcting codes with redundancy 3n-2 for infinitely many prime lengths, and it proves that this family exists unconditionally by allowing the cyclic slope to vary among 2, -3, and -15. The graph Singleton bound for three erasures is 3n-3, so these constructions are one redundancy bit above optimal.

The paper also gives Singleton-optimal triple-node codes at the finite lengths n=6, 8, 10, and 12 through an ordinary-edge Moore skeleton and a loop-completion procedure. These finite examples show that the extra bit is not inherent to the graph-erasure model. The unresolved issue is whether optimal triple-node codes exist for infinitely many lengths.