2317 — Six Bits Are Minimally Sufficient for a Protected D(S3) Sector Record
Verdict
The raw three-bit record uses every word of F2^3; all 24 single-bit flips
therefore produce another valid sector label. A one-error-correcting binary
record for eight sectors must obey 8(1+n) <= 2^n, ruling out every
n <= 5. Puncturing the length-seven binary simplex code gives an exact
[6,3,3] code, so six bits are necessary and sufficient. All 48 one-bit
corruptions decode uniquely.
Scope
The code protects a final record after the correct label reaches the encoder. It does not repair controlled-power, inverse-Fourier, wrong-input, or unprotected encoder faults, and it does not supply the missing controlled unitaries.
Durable verification
- Packet:
research/kitaev/s3-sector-record-minimal-redundancy.md - Checker:
python research/kitaev/checkers/check_s3_sector_record_redundancy.py - Result:
research/kitaev/results/s3-sector-record-redundancy.json - Epistemic graph:
ev-000000003183-b0b8ea16-56d6-4338-9899-b8f8d434f8d2 - Ledger allocation:
seqclaim-939790d48ea19651b7e246ff