A size-30 constant-weight code with parameters n=26, d=8, w=5

Construction and verification

The prescribed action was Z_24 + 2 fixed, of order 24, with group specification {"name":"Z_24 + 2 fixed","gens":[{"op":"cycle","from":0,"len":24}]}. The group acts on the 5-subsets of the ground set of size 26. The construction was restricted to unions of complete block orbits under this action.

Each block orbit was first tested internally. An orbit was discarded if any pair of its members had intersection exceeding lambda=1. The surviving orbits were used as vertices of a compatibility graph. Two vertices were adjacent exactly when every block in either orbit was compatible with every block in the other orbit. Each vertex was weighted by its orbit length. A maximum-weight clique therefore gives a largest code invariant under the prescribed group.

This optimization produced and exhaustively established an orbit-union maximum of 30 for Z_24 + 2 fixed. The selected union was then checked independently by scanning all pairs of blocks and testing the intersection cap directly, without using orbit representatives, orbit compatibility, or the clique computation. This scan verified size 30 and pairwise intersection at most 1.

The Schonheim upper bound is 31, leaving a gap of 1. The gap is not closed: the exhaustive computation proves only that Z_24 + 2 fixed cannot support a larger invariant orbit-union code. It does not exclude a code of size 31 without that symmetry.

Search context and conjecture

On this cell, 148 groups were tried, and 147 produced a code. Orbit-union maxima were determined exhaustively for 143 groups. Named exhaustive outcomes included Z_26, of order 26, with maximum 26; Z_25 + 1 fixed, of order 25, with maximum 25; Z_25:20 under x -> 2x + 1 fixed and under x -> 3x + 1 fixed, each of order 500, with maximum 5; Z_24 + 2 fixed, of order 24, with maximum 30; and Z_24:2 under x -> 5x + 2 fixed, of order 48, with maximum 30.

Further named outcomes were Z_23 + 3 fixed, of order 23, with maximum 23; Z_23:11 under x -> 2x + 3 fixed and under x -> 3x + 3 fixed, each of order 253, with maximum 0; Z_23:22 under x -> 5x + 3 fixed, of order 506, with maximum 0; Z_13 x Z_2 blocks, of order 169, with maximum 0; and Z_21 + 5 fixed, of order 21, with maximum 22. These values bound only codes invariant under the named groups, not arbitrary codes.

A search-derived conjecture is that, at lambda=1, successful actions require internally admissible short block orbits whose lengths combine effectively. Z_24 + 2 fixed permits lengths 24+6, while low-order F5^2 actions provide orbit sizes 5, 10, 20, or 25. In contrast, full Z_26 appears effectively restricted to size 26, Z_25 cannot represent 31, and multiplier groups of orders 125-500 may fuse blocks into coarse or self-colliding orbits. Also conjecturally, 2 globally fixed points in a moving block force a forbidden intersection of at least 2.

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) = (26,8,5) 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 + 2 fixed, specified as {"name":"Z_24 + 2 fixed","gens":[{"op":"cycle","from":0,"len":24}]}. Schonheim upper bound for these parameters: 31.

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 25
1 7 13 19 25
2 8 14 20 25
3 9 15 21 25
4 10 16 22 25
5 11 17 23 25

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