DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Bilinear dual of a finite-field linear code

Definition

For $x, y \in F^{n}$ , define the standard bilinear dot product $x \cdot y : = \sum_{j < n} x_{j} y_{j}$ . For any linear subspace $C \leq F^{n}$ (Linear subspace of a vector space)—in particular, the codeword subspace of a classical linear code as in Classical linear code over a finite field—its dual code is

$C^{⊥} : = {y \in F^{n} : x \cdot y = 0 for every x \in C} .$

This is a bilinear pairing over $F$ ; it is not a positive-definite or Hermitian inner product. The proof that $C^{⊥}$ is a linear subspace, has dimension $n - k$ , and satisfies $(C^{⊥})^{⊥} = C$ is Dimension and double-dual identity for finite-field code duals.

Depends on

Used by

CSS code from nested binary codes Definition
Generator and parity-check matrices over a finite field Definition
CSS encoding from nested binary codes Example
Dimension of the binary dual code Lemma
CSS encoding dimension and commutation Theorem
CSS syndrome correction Theorem
Dimension and double-dual identity for finite-field code duals Theorem
Every finite-field linear code has generator and parity-check matrices Theorem

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard Cleve, Quantum Information Processing course notes, §37.1.1 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 7, §7.6 (standard reference, not scraped)