Generator matrix over GF(16)
| Polynomial | g₀ = 1 | g₁ = α | g₂ = α² | g₃ = α³ |
|---|
01 · Change the metric
One vector, two different weights
Each symbol belongs to GF(2⁴) = GF(2)[α]/(α⁴ + α + 1). Hamming weight counts nonzero extension-field symbols. Rank weight expands each symbol in the ordered basis (1, α, α², α³) and takes the binary matrix rank.
Current computation
- Hamming weight
- Rank weight
| Basis row | v₀ | v₁ | v₂ | v₃ |
|---|
02 · Build the code
A complete [4, 2, 3] Gabidulin census
For message (a₀,a₁), the encoder evaluates the linearized polynomial f(x)=a₀x+a₁x² at g=(1,α,α²,α³). The four evaluation points are linearly independent over GF(2). Every figure below is recomputed from all 256 messages.
Length4
Dimension2
Minimum rank distance
Unique radius
Exhaustive evidence
- Messages / distinct words
- Nonzero rank distribution
- Measured minimum distance
- Singleton bound
03 · Cross the guarantee
Decode by actual rank distance
This finite decoder checks every one of the 256 codewords and returns a word only when exactly one lies at rank distance at most t=1. It is an auditable exhaustive decoder—not Gabidulin’s efficient algebraic algorithm.
Decoder verdict
- Transmitted word
- Error vector
- Error weights
- Received word
- Reported message
- Independent check
Complete beyond-radius control
Enumerate every rank-two error from the zero word
This checks the entire finite error set, not a random sample. “Correct” means recovering the transmitted zero message; “miscorrected” means returning a different codeword within the decoder’s radius.
No census has run in this page session.
04 · Keep the claim bounded
Code family is not deployment status
| Name | What this lesson establishes | What it does not establish |
|---|---|---|
| Gabidulin code | Executable MRD code mathematics in a 256-word teaching space. | No encryption, KEM, security reduction, production parameters, or efficient decoder. |
| GPT / Faure–Loidreau | Named as historical rank-code public-key constructions. | Not implemented; this code demo does not transfer security to them. Published structural attacks must be assessed per construction. |
| RQC | Named as a structured rank-metric KEM considered by NIST. | Not implemented or endorsed. NIST IR 8309 records that its Round-2 parameter sets were reduced below 128-bit security and it did not advance. |
| Classic McEliece / HQC | Useful comparison labs in the same broad code-based family. | They use Hamming-metric constructions; they are not Gabidulin or rank-metric schemes. |
Primary and authoritative sources
- Gabidulin, “Theory of Codes with Maximum Rank Distance” (1985) — MRD construction and decoding.
- Gaborit, Otmani, and Talé Kalachi, arXiv:1606.07760 — rank-weight definition, Gabidulin generator form, and a construction-specific structural attack.
- NIST IR 8309, §3.16 — RQC’s 2020 third-round selection status and cryptanalytic record.
Secondary orientation: Gabidulin, “Schemes Based on Rank Codes,” in the Encyclopedia of Cryptography, Security and Privacy, 3rd ed. (2025), DOI entry 388.