Algebraic geometry codes from Hermitian curves over finite fields. Exceed Gilbert-Varshamov bound. Defined over x^q + y^q + 1 = 0 in GF(q²). Parameters [n=q³, k, d] where n = q³ is the number of rational points. Used in deep space communications and coding theory research. Achieve better rates than Reed-Solomon codes.
| Property | Value |
|---|---|
| Category | Error Correction |
| Sub-category | Algebraic Geometry Code |
| Security status | 🎓 Educational Only |
| Complexity | Expert |
| Inventor | V. D. Goppa, Garcia-Stichtenoth |
| Year | 1981 |
| Origin | 🇷🇺 Russia |
| Source | algorithms/ecc/hermitian-code.js |
Status: 🎓 Educational Only
| Issue | Description | Mitigation |
|---|---|---|
| Decoding Complexity | Algebraic geometry decoding algorithms are computationally intensive compared to Reed-Solomon. | — |
| Field Size Requirements | Requires large finite fields for practical parameters - field size must be square for Hermitian curve. | — |
| Construction Complexity | Curve theory and rational points computation requires advanced algebraic geometry knowledge. | — |
6 vectors ship with this algorithm and run in the test suite. Byte values are hexadecimal.
Vector 1 — Hermitian [8,3] all zeros codeword
| Field | Value |
|---|---|
input |
000000 |
expected |
0000000000000000 |
Vector 2 — Hermitian [8,3] basis function f=1 (constant)
| Field | Value |
|---|---|
input |
010000 |
expected |
0101010101010101 |
Vector 3 — Hermitian [8,3] basis function f=x
| Field | Value |
|---|---|
input |
000100 |
expected |
0001020302030001 |
Vector 4 — Hermitian [8,3] basis function f=y
| Field | Value |
|---|---|
input |
000001 |
expected |
0002010303010200 |
Vector 5 — Hermitian [8,3] linear combination f=1+x
| Field | Value |
|---|---|
input |
010100 |
expected |
0100030203020100 |
Vector 6 — Hermitian [8,3] linear combination f=x+y
| Field | Value |
|---|---|
input |
000101 |
expected |
0003030001020201 |