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Tensor Product Preserves No-Zero Coordinates

Claim/Theorem

Let \(A\subseteq \mathbf F_2^m\) and \(B\subseteq \mathbf F_2^n\) be binary linear codes. If both \(A\) and \(B\) have no zero coordinates, then the tensor-product code

\[ A\otimes B\subseteq \mathbf F_2^{m\times n} \]

also has no zero coordinates.

Indeed, for any coordinate pair \((u,v)\), choose \(a\in A\) with \(a_u=1\) and \(b\in B\) with \(b_v=1\). Then \((a\otimes b)_{u,v}=1\), so the coordinate \((u,v)\) is not identically zero on \(A\otimes B\).

Combined with [[dual-distance-excludes-zero-coordinates.md]], this yields the Quantum-Tanner-specific corollary:

if

\[ d(C_A),\ d(C_B),\ d(C_A^\perp),\ d(C_B^\perp)\ \ge\ 2, \]

then both local tensor codes

\[ C_A\otimes C_B \qquad\text{and}\qquad C_A^\perp\otimes C_B^\perp \]

have no zero coordinates. Therefore both parity gadgets in the chosen stabilizer presentation admit connected local bases via [[connected-basis-for-nonzero-coordinate-code.md]].

Dependencies

  • [[dual-distance-excludes-zero-coordinates.md]]
  • [[connected-basis-for-nonzero-coordinate-code.md]]

Conflicts/Gaps

  • This node only proves the no-zero-coordinate property for the local tensor codes. It does not yet show expansion of the global Tanner graph.
  • The argument is deterministic once the factor-code distance conditions hold, but it does not provide an explicit construction of local codes satisfying those distance conditions. That remains the role of Theorem 18 in the source paper.

Sources

  • 10.48550/arXiv.2202.13641