ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-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.

Binary three-bit repetition code and its syndrome table

Example

Work over $F = F_{2} = Z / 2 Z$ . Let $C = {000, 111} \leq F^{3}$ , with column-vector generator and parity-check matrices

$G = [\begin{array}{c} 1 \\ 1 \\ 1 \end{array}], H = [\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 1 \end{array}] .$

The code has parameters $[3, 1, 3]_{2}$ . The four error vectors of weight at most one have distinct syndromes $H e$ , giving a unique correction table.

Facts & Assumptions

Given: The field $F = Z / 2 Z$ , code $C = {000, 111}$ , and matrices $G, H$ displayed above.

[F1]

For every prime $p$ , the two operations on $Z / p$ make it a field: For every prime $p$ , $Z / p$ with its usual residue-class operations is a field.

[L1]

Generator and parity-check matrices over a finite field: A generator matrix has independent columns spanning $C$ ; a parity-check matrix has independent rows and nullspace $C$ ; and with column-vector conventions their product is zero.

[L2]

Syndrome of a received word over a finite field: The syndrome of a received column $r$ is $H r$ .

[L3]

Classical distance gives error detection, unique decoding, and erasure recovery: A code of distance $d$ uniquely decodes up to $⌊ (d - 1) / 2 ⌋$ unknown errors.

Verification

1.1F1L1givenalgebra

The prime $2$ makes $F$ a field by [F1]. The single nonzero column of $G$ is independent and spans ${000, 111}$ . Multiplication gives $H G = [\begin{array}{c} 0 \\ 0 \end{array}]$ . Solving $H x = 0$ gives $x_{0} = x_{1} = x_{2}$ , so $\ker H = C$ . If $a, b \in F_{2}$ and $a (1, 1, 0) + b (0, 1, 1) = 0$ , coordinate $0$ gives $a = 0$ and coordinate $2$ gives $b = 0$ ; hence the rows of $H$ are independent. Thus $G$ and $H$ meet the generator and parity-check definitions [L1].

1.2givenL1algebra

The only nonzero codeword is $111$ , whose weight is $3$ ; hence $k = 1$ and $d = 3$ .

1.3L2givenalgebra

Direct multiplication over $F_{2}$ gives $H (000) = 00$ , $H (100) = 10$ , $H (010) = 11$ , and $H (001) = 01$ . These are all four errors of weight at most one, and their syndromes are distinct.

2.1step 1.3L2L3given∎

For a received word $r = c + e$ with $c \in C$ and $wt (e) \leq 1$ , step 1.3 identifies $e$ from $H r = H e$ ; subtracting it yields $r - e = c$ . This realizes the one-error uniqueness guarantee of [L3].

Remarks

The matrix orientation is column-vector orientation: received words are columns and the syndrome is $H r \in F_{2}^{2}$ . The source notes use row-vector conventions in some passages; the displayed calculations are recomputed in the root library convention.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Richard Cleve, Quantum Information Processing course notes, §§36.1 and 37.1–37.1.3 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 7, §7.6 (standard reference, not scraped)