Good Code Parameters Do Not Imply Cut Rank

Claim/Theorem

Good code parameters alone do not force large intrinsic cross-cut rank.

More precisely, let \(C^{(1)}_n\) and \(C^{(2)}_n\) be two families of linear codes of equal blocklength \(N\) with

\[ k_i=\Theta(N), \qquad d_i=\Theta(N). \]

Form the direct-sum family

\[ C_n \;=\; C^{(1)}_N \oplus C^{(2)}_N \]

on disjoint coordinate sets of total length \(n=2N\). Then

\[ k(C_n)=k_1+k_2=\Theta(n), \qquad d(C_n)=\min(d_1,d_2)=\Theta(n), \]

so the family remains asymptotically good. However, for the balanced partition \(L\) equal to the coordinates of the first summand,

\[ \dim(C_n|_L)=k_1, \qquad \dim(C_n|_{L^c})=k_2, \qquad \dim(C_n)=k_1+k_2, \]

and therefore [[cross-cut-stabilizer-rank-rank-formula.md]] gives

\[ \lambda_{C_n}(L) \;:=\; \dim(C_n|_L)+\dim(C_n|_{L^c})-\dim(C_n) \;=\;0. \]

Equivalently, for the corresponding direct-sum stabilizer spaces, the intrinsic cross-cut stabilizer rank satisfies

\[ \chi_L(\mathcal S_n)=0 \]

across the same balanced cut.

Thus no theorem that uses only the global parameters [n,k,d] can prove the linear balanced-cut rank-connectivity required by the current generator-invariant Conjecture-3 frontier. Any successful \chi_L lower bound must exploit additional irreducibility structure such as Tanner expansion, connectedness of the presentation, local testability, or some stronger form of matroid connectivity.

Dependencies

• [[cross-cut-stabilizer-rank-rank-formula.md]]

Conflicts/Gaps

• This is a negative structural example, not a positive lower bound.
• The counterexample uses disconnected direct sums. It does not show that connected expander-style QLDPC families fail to have linear balanced-cut connectivity.
• The node explains why the presentation-invariant 2D theorem from [[quantum-tanner-good-family-presentation-invariant-2d-barrier.md]] does not by itself solve the stronger cut-rank frontier.

Sources

• 10.48550/arXiv.2109.14599
• 10.48550/arXiv.0805.2199