A size-30 constant-weight code with parameters n=27, d=8, w=5

Construction and verification

The construction has parameters n=27, d=8, w=5 and size 30. Equivalently, it is a family of 30 distinct 5-subsets of a 27-set such that every pair intersects in at most lambda=1.

The prescribed automorphism group was Z_24 + 3 fixed, of order 24, with specification {"name":"Z_24 + 3 fixed","gens":[{"op":"cycle","from":0,"len":24}]}. The group acts on the 5-subsets. Each orbit was first tested internally, and any orbit containing members whose intersection exceeded 1 was discarded. The remaining orbits were represented as vertices of a compatibility graph. Two vertices were compatible when every block in one orbit met every block in the other orbit in at most 1 point. Vertex weights were the orbit sizes. A maximum-weight clique therefore gave a largest code invariant under the prescribed group.

For Z_24 + 3 fixed, the exhaustive orbit-union maximum was 30, and the reported construction attains it. After construction, the complete family was checked independently by a pairwise scan using none of the orbit generation, orbit filtering, compatibility-graph, or clique machinery. This scan verified size 30, constant weight 5, and pairwise intersection at most 1, hence minimum distance 8.

The Schonheim upper bound is 32, so the gap is 2 and is not closed. The group-restricted exhaustive result does not prove that 30 is the unrestricted maximum for these parameters.

Search context and conjecture

On this cell, 146 groups were tried and 145 produced a code. Orbit-union maxima were determined exhaustively for 140 groups. Named exhaustive results include: Z_27, order 27, maximum 27; Z_26 + 1 fixed, order 26, maximum 26; Z_26:3 with x -> 3x and 1 fixed, order 78, maximum 0; Z_26:4 with x -> 5x and 1 fixed, order 104, maximum 0; Z_25 + 2 fixed, order 25, maximum 25; both listed Z_25:20 actions, of order 500, maximum 5; Z_24 + 3 fixed, order 24, maximum 30; Z_24:2 with x -> 5x and 3 fixed, order 48, maximum 30; Z14 + 13 fixed, order 14, maximum 3; Z15 + 12 fixed, order 15, maximum 6; and Z16 + swap + 9 fixed, order 32, maximum 6. These maxima bound only codes invariant under the named groups, not arbitrary codes.

Conjecture from the search: successful prescriptions tend to be small, nearly semiregular diagonal groups with few global fixed points and orbit sizes able to sum to 30 or 32. Large groups appear to force exceptional stabilizers, while long cycles and independent-factor actions restrict usable orbit structure. Among regular order-8 actions, Q8 appeared preferable to D8, C4 x C2, and C2^3 because its single involution yielded fewer repeated group differences. Many global fixed points also appeared harmful. These statements are search conjectures, not verified general results. Whether the construction improves published values is for reviewers to assess.

Verification

The code below is the complete witness: 30 codewords, one per line, as ascending 0-based positions. It is valid for (n,d,w) = (27,8,5) if and only if every line has exactly 5 entries, no line repeats, and every pair of lines shares at most 1 entries. That is a pairwise scan and requires nothing from the construction above.

Prescribed group: Z_24 + 3 fixed, specified as {"name":"Z_24 + 3 fixed","gens":[{"op":"cycle","from":0,"len":24}]}. Schonheim upper bound for these parameters: 32.

0 4 5 7 15
1 5 6 8 16
0 9 13 14 16
2 6 7 9 17
1 10 14 15 17
3 7 8 10 18
2 11 15 16 18
4 8 9 11 19
3 12 16 17 19
0 1 3 11 20
5 9 10 12 20
4 13 17 18 20
1 2 4 12 21
6 10 11 13 21
5 14 18 19 21
2 3 5 13 22
7 11 12 14 22
6 15 19 20 22
0 8 17 21 22
3 4 6 14 23
8 12 13 15 23
0 2 10 19 23
7 16 20 21 23
1 9 18 22 23
0 6 12 18 26
1 7 13 19 26
2 8 14 20 26
3 9 15 21 26
4 10 16 22 26
5 11 17 23 26

This block is generated mechanically from the verified search record and is not model output.