A 35-word constant-weight code with parameters n=29, d=8, w=5

Construction and verification

The prescribed automorphism group is Z_28 + 1 fixed, of order 28, specified by {"name":"Z_28 + 1 fixed","gens":[{"op":"cycle","from":0,"len":28}]}. The group acts on the 5-subsets of the 29-set. The code was required to be a union of orbits under this action.

Each block orbit was first tested internally. An orbit was discarded if any two of its members had intersection greater than 1. The surviving orbits became vertices of a compatibility graph. Two vertices were adjacent exactly when every block in one orbit had intersection at most 1 with every block in the other. Each vertex was weighted by its orbit length. A maximum-weight clique therefore gives a largest code invariant under the prescribed group.

This procedure produced an invariant code of size 35. After construction, the resulting 35 blocks were verified independently by scanning every pair and checking the intersection cap directly, without using orbit information or the compatibility graph. Thus the appended construction can be re-run as a family of 5-subsets and checked solely by pairwise intersections.

The Schonheim upper bound is 40, so the verified construction has gap 5. This gap is not closed. Whether size 35 improves on published values is for reviewers to assess.

Group-restricted results and search conjecture

On this cell, 152 groups were tried, and 148 produced a code. Orbit-union maxima were determined exhaustively for 143 groups. These exhaustive values bound only codes invariant under the named groups; they do not bound unrestricted constant-weight codes.

The reported exhaustive maxima include 29 for Z_29 of order 29, 35 for Z_28 + 1 fixed of order 28, and 27 for Z_27 + 2 fixed of order 27. Z_26 + 3 fixed, of order 26, has maximum 26. The two listed Z_28:6 actions with 1 fixed point, each of order 168, have maximum 7. The two listed Z_27:18 actions with 2 fixed points, each of order 486, have maximum 0. Z_26:3 with 3 fixed points, of order 78, and Z_26:4 with 3 fixed points, of order 104, also have maximum 0. C20 x C2 with 7 fixed points, of order 40, has maximum 11; C15 x C2 with 12 fixed points, of order 30, has maximum 7.

Conjecture from the search: useful actions tend to have modest order, few fixed points, and compatible orbit lengths that combine flexibly. For Z_28 + 1 fixed, usable lengths include 28 and 7, which combine to 35. Large multiplier groups often create too many translates, causing a 5-set to intersect one of its translates in more than 1 point. Many fixed points can also obstruct moving blocks containing two fixed points. Order alone does not determine performance, since reflections may invalidate otherwise usable difference-type orbits. A further conjectural leave-based heuristic favors actions preserving a suitable 8-point union of point-orbits and offering orbit lengths that can sum to 39.

Verification

The code below is the complete witness: 35 codewords, one per line, as ascending 0-based positions. It is valid for (n,d,w) = (29,8,5) 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_28 + 1 fixed, specified as {"name":"Z_28 + 1 fixed","gens":[{"op":"cycle","from":0,"len":28}]}. Schonheim upper bound for these parameters: 40.

0 1 3 9 13
1 2 4 10 14
2 3 5 11 15
3 4 6 12 16
4 5 7 13 17
5 6 8 14 18
6 7 9 15 19
7 8 10 16 20
8 9 11 17 21
9 10 12 18 22
0 4 19 20 22
10 11 13 19 23
1 5 20 21 23
0 15 16 18 24
11 12 14 20 24
2 6 21 22 24
1 16 17 19 25
12 13 15 21 25
3 7 22 23 25
2 17 18 20 26
13 14 16 22 26
4 8 23 24 26
0 6 10 25 26
0 2 8 12 27
3 18 19 21 27
14 15 17 23 27
5 9 24 25 27
1 7 11 26 27
0 7 14 21 28
1 8 15 22 28
2 9 16 23 28
3 10 17 24 28
4 11 18 25 28
5 12 19 26 28
6 13 20 27 28

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