Linear codes

[87,8,63] over GF(7)

A [87,8] linear code over GF(7) with minimum distance 63, one more than the 62 listed at codetables.de, where the upper bound is 68. 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 = 63
Previous best	d = 62 (upper bound 68)
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 5,764,801 codewords differ in at least 63 of the 87 positions, so it corrects up to 31 errors. It also gives [86,8,62] (the table lists 61), by deleting one coordinate and [85,8,61] (the table lists 60), by deleting two coordinates.

How it was found and checked

A quasi-cyclic code with 3 circulant blocks of size 29, 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 [87,8] code over GF(7)?

Ours, with minimum distance 63; codetables.de listed 62 (read 26 October 2026).

How was it found?

As a quasi-cyclic code with 3 circulant blocks of size 29, by a local search that checks promising candidates exactly.

Is it on codetables.de yet?