Optimal-rate construction for non-bipartite graph codes

Construct linear [N, δ]_q-graph codes over symmetric N × N matrices with zero diagonals (i.e., codes that recover from erasing all rows and columns indexed by any set of at most δN vertices) that achieve rate at least (1 − δ)^2 − o(1) as N grows, thereby attaining the capacity for non-bipartite graph codes.

Background

The paper defines [N, δ]_q-graph codes as linear codes over symmetric N × N matrices (with zero diagonals) that can recover from erasures of all rows and columns corresponding to any set of up to δN vertices. It is known that the capacity of this model is (1 − δ)^2, and random linear codes achieve this rate asymptotically.

The authors give strongly explicit constructions that achieve the optimal trade-off for the bipartite (matrix) variant and provide improved rates for non-bipartite graph codes, specifically achieving rate (1 − √δ)^4 − o(1). However, reaching the optimal (1 − δ)^2 − o(1) rate for non-bipartite graph codes remains unresolved.