LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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.

A received-word syndrome depends only on the error vector

Statement

For a codeword $c \in C$ , an error $e \in F^{n}$ , and received word $r = c + e$ , the syndrome satisfies $s (r) = H e$ .

Facts & Assumptions

Given: A codeword $c \in C$ , error vector $e \in F^{n}$ , received word $r = c + e$ , and a parity-check matrix $H$ for $C$ .

[L1]

Syndrome of a received word over a finite field: The syndrome map is $s (r) = H r$ for each received word $r$ .

[L2]

Syndrome of a received word over a finite field: Its hypothesis makes $H$ a parity-check matrix of $C$ ; the matrix convention in that definition's prerequisites is $\ker H = C$ , hence $H c = 0$ for every codeword.

Proof

1.1L1givenalgebra

By [L1] and the linearity of matrix multiplication, $s (r) = H (c + e) = H c + H e$ .

2.1L2step 1.1∎

Since $c \in C = \ker H$ by [L2], $H c = 0$ , so step 1.1 gives $s (r) = H e$ . The value depends on $e$ and not on which codeword was transmitted.

Depends on

Used by

Classical distance gives error detection, unique decoding, and erasure recovery Theorem

Dependency tree · two levels

10 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, §37.1.3 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 7, §7.6 (standard reference, not scraped)