Balanced triples attaining finite-degree GV

Prove that for every balanced triple of degrees (j_Z, j_X, k) satisfying 4 ≤ j_Z < k/2 and j_X = k − j_Z, the two classical constituent codes in the nested regular sparse construction—namely the HA-side code C_Z = B(Ker A_Z) and the MN-side code C_X defined by C_X = { x ∈ F_2^n : ∃ u ∈ F_2^{m_X} with A_X^T u + B^T x = 0 }, where A_X = [A_Z; A_Δ] with column degrees j_Z and j_Δ = j_X − j_Z and common row degree k, and B is a square (k,k)-regular sparse map—attain the classical Gilbert–Varshamov distance already at fixed finite degree; consequently, establish that the associated nested CSS code family attains the CSS Gilbert–Varshamov distance at finite degree.

Background

The paper constructs a nested CSS code family from regular LDPC matrices A_Z, A_Δ, and a square (k,k)-regular map B, forming A_X = [A_Z; A_Δ]. Under balanced triples (j_Z, j_X, k) with j_X = k − j_Z, the authors prove fixed-degree linear distance on both sides and rigorously certify finite-degree attainment of the classical GV distance for seven explicit triples.

Based on numerical evidence and rigorous certifications for these examples, the authors conjecture that all balanced triples with 4 ≤ j_Z < k/2 also achieve the classical GV distance at finite degree on both the HA and MN sides, which would imply finite-degree attainment of the CSS Gilbert–Varshamov bound for the resulting nested CSS codes.