Linear codes
[39,12,19] over GF(5)
A [39,12] linear code over GF(5) with minimum distance 19, one more than the 18 listed at codetables.de, where the upper bound is 22. Quantavane found it on 24 October 2026.
Sent to Markus Grassl for codetables.de on 24 October 2026; the table shows it once he adds it.
The record
| Ours | d = 19 |
|---|---|
| Previous best | d = 18 (upper bound 22) |
| Previous source | the lower bound listed at codetables.de |
| Found | 24 October 2026 |
| Status | Sent to Markus Grassl for codetables.de on 24 October 2026; the table shows it once he adds it. |
| Check | Two independent programs computed the minimum distance; one of them enumerates every codeword. |
Any two of its 244,140,625 codewords differ in at least 19 of the 39 positions, so it corrects up to 9 errors.
How it was found and checked
A quasi-cyclic code with 3 circulant blocks of size 13, found by a local search over quasi-twisted codes that scores candidates by their lightest sparse codewords and computes the exact distance only for promising ones. Two independent programs computed the minimum distance; one of them enumerates every codeword.
A linear code adds redundancy to data so that errors can be corrected. An [n,k] code over the field GF(q) turns k symbols into n. Its minimum distance d is the fewest positions in which two codewords differ, and the code corrects up to (d−1)/2 errors. Markus Grassl's tables at codetables.de keep the largest d anyone has reached for each n and k. These codes go one higher.
Questions
What is the best known [39,12] code over GF(5)?
Ours, with minimum distance 19; codetables.de listed 18 (read 26 October 2026).
How was it found?
As a quasi-cyclic code with 3 circulant blocks of size 13, by a local search that checks promising candidates exactly.
Is it on codetables.de yet?
Sent to Markus Grassl for codetables.de on 24 October 2026; the table shows it once he adds it.