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Dual Distance Excludes Zero Coordinates

Claim/Theorem

This node is a graph-internal coding-theory lemma used to simplify the explicit Quantum Tanner route. Let \(C\subseteq \mathbf F_2^N\) be a binary linear code, and let \(e_u\) denote the standard basis vector at coordinate \(u\).

Then the following are equivalent for a fixed coordinate \(u\):

  1. every codeword of \(C\) is zero at coordinate \(u\),
  2. \(e_u\in C^\perp\).

Consequently:

  • \(C\) has no zero coordinates if and only if \(C^\perp\) contains no weight-1 vector,
  • equivalently,
\[ C\text{ has no zero coordinates}\quad\Longleftrightarrow\quad d(C^\perp)\ge 2. \]

By symmetry,

\[ C^\perp\text{ has no zero coordinates}\quad\Longleftrightarrow\quad d(C)\ge 2. \]

So in any setting where both \(d(C)\ge 2\) and \(d(C^\perp)\ge 2\), both \(C\) and its dual automatically satisfy the no-zero-coordinate condition needed by [[connected-basis-for-nonzero-coordinate-code.md]].

Dependencies

  • [[connected-basis-for-nonzero-coordinate-code.md]]

Conflicts/Gaps

  • This lemma only removes the zero-coordinate obstruction. It does not say anything about robustness, local testability, or expansion.
  • The statement is linear-algebraic and does not by itself identify which concrete local codes in the quantum Tanner construction satisfy the hypothesis. That is handled by [[tensor-product-preserves-no-zero-coordinates.md]] and [[quantum-tanner-theorem17-parity-expander.md]].

Sources

  • 10.48550/arXiv.2202.13641