Results
Linear code records
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.
They come from a local search over quasi-cyclic and quasi-twisted codes that scores candidates by their lightest sparse codewords and computes the exact minimum distance only for promising ones. Each distance was computed by two independent programs.
9 sent to its maintainer and 3 waiting for our next batch.
Twelve linear codes
| Entry | Ours | Previous best | Status |
|---|---|---|---|
| [220,13,125] over GF(3) | 125 | 124 | sent |
| [153,10,99] over GF(4) | 99 | 98 | next batch |
| [39,12,19] over GF(5) | 19 | 18 | sent |
| [70,12,40] over GF(5) | 40 | 39 | sent |
| [93,13,55] over GF(5) | 55 | 54 | sent |
| [57,10,36] over GF(7) | 36 | 35 | next batch |
| [66,12,40] over GF(7) | 40 | 39 | sent |
| [87,8,63] over GF(7) | 63 | 62 | sent |
| [90,11,59] over GF(7) | 59 | 58 | next batch |
| [102,10,73] over GF(9) | 73 | 72 | sent |
| [130,10,96] over GF(9) | 96 | 95 | sent |
| [130,13,90] over GF(9) | 90 | 89 | sent |
Questions
What does [n,k,d] mean?
A linear code of length n and dimension k whose minimum distance is d: any two codewords differ in at least d positions, so the code corrects up to (d−1)/2 errors.
Where are the best known codes listed?
In Markus Grassl's tables at codetables.de, which give the best known lower bound and the proven upper bound on d for each length and dimension over small fields.
When will codetables.de show them?
When its maintainer adds them. We send codes in batches; each page here says when it was sent.