Minimum distance of parameterized codes for broad classes of graphs

Derive explicit formulas for the minimum distance of the parameterized codes C_X(d) associated with simple graphs G beyond the cases where X is the projective torus and G is complete bipartite, covering broader graph families and parameter ranges.

Background

The minimum distance is a central parameter in coding theory. A formula is known when X is the projective torus and d is less than the regularity, which also yields the case of complete bipartite graphs. Outside these cases, the minimum distance had largely remained unknown. This paper contributes by computing the minimum distance for codes of order 1 over even cycles, but the general problem across other graph families and code orders remains unresolved.

The authors explicitly note the lack of known formulas for other classes of graphs, emphasizing the broader open question of determining minimum distances for parameterized codes in general settings.