Cyclic codes constructed from quadratic residues in finite fields. For prime p ≡ ±1 (mod 8), constructs (p, (p+1)/2) code with excellent distance properties. Binary Golay code is a famous QR code. Automorphism group includes field automorphisms. Used in deep space and satellite communications.
| Property | Value |
|---|---|
| Category | Error Correction |
| Sub-category | Cyclic Code |
| Security status | 🎓 Educational Only |
| Complexity | Advanced |
| Inventor | Andrew Gleason, Solomon Golomb |
| Year | 1958 |
| Origin | 🇺🇸 United States |
| Source | algorithms/ecc/quadratic-residue-code.js |
Status: 🎓 Educational Only
| Issue | Description | Mitigation |
|---|---|---|
| Limited Code Lengths | Only defined for specific prime lengths p ≡ ±1 (mod 8), limiting flexibility. | — |
| Complex Decoding | Optimal decoding requires algebraic techniques more complex than simple codes. | — |
4 vectors ship with this algorithm and run in the test suite. Byte values are hexadecimal.
Vector 1 — QR (7,4) all zeros
| Field | Value |
|---|---|
input |
00000000 |
expected |
00000000000000 |
Vector 2 — QR (7,4) pattern 1000
| Field | Value |
|---|---|
input |
01000000 |
expected |
01000000010100 |
Vector 3 — QR (7,4) pattern 0100
| Field | Value |
|---|---|
input |
00010000 |
expected |
00010000000101 |
Vector 4 — QR (7,4) all ones
| Field | Value |
|---|---|
input |
01010101 |
expected |
01010101010101 |