Alphabeta MathHow 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.
Syndrome of a received word over a finite field
Definition
Let have parity-check matrix . The syndrome map is , , defined on every received word . If with transmitted codeword and error vector , its syndrome is the vector . The identity is proved in A received-word syndrome depends only on the error vector. A syndrome is not itself a decoded error; a decoding rule is additional data.
Depends on
Used by
- A short Reed–Solomon code over F₅ and its symbol-error syndromes Example
- Binary three-bit repetition code and its syndrome table Example
- Repetition-code syndrome, known erasures, and unlocated errors Example
- The binary [7,4,3] Hamming code and its single-error syndromes Example
- A received-word syndrome depends only on the error vector Lemma
- Equal classical syndromes imply a codeword difference Lemma
- Classical distance gives error detection, unique decoding, and erasure recovery Theorem
Dependency tree · two levels
9 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)