Enlarge a constant-weight binary code on a shipped cell list
OpenMathematics Statistics
No external cash prize - leaderboard standing and recognition only.
A peer-reviewed paper exhibiting, for at least one cell (n,d,w) on the list below, an explicit constant-weight binary code together with its size s, verifiable by direct computation from the paper alone. THE OBJECT. A code is a set of w-element subsets of {0,...,n-1}, written one codeword per line as ascending 0-based positions separated by spaces. It is VALID for (n,d,w) if and only if: every codeword has exactly w elements; no codeword is repeated; and every pair of distinct codewords A,B satisfies |A intersect B| <= w - d/2, which is equivalent to Hamming distance at least d. Checking validity is a pairwise scan and needs no theory. THE CELLS, each with its Schonheim upper bound U. Every U here is computed from the formula below, not cited from any table: n=17 d=6 w=4 lambda=1 U=21 n=18 d=6 w=4 lambda=1 U=22 n=19 d=6 w=4 lambda=1 U=28 n=20 d=6 w=4 lambda=1 U=30 n=21 d=6 w=4 lambda=1 U=31 n=28 d=6 w=4 lambda=1 U=63 n=22 d=8 w=5 lambda=1 U=22 n=26 d=8 w=5 lambda=1 U=31 n=27 d=8 w=5 lambda=1 U=32 n=28 d=8 w=5 lambda=1 U=33 n=29 d=8 w=5 lambda=1 U=40 n=30 d=8 w=5 lambda=1 U=42 n=22 d=8 w=6 lambda=2 U=77 n=24 d=8 w=6 lambda=2 U=92 n=25 d=8 w=6 lambda=2 U=100 n=26 d=10 w=6 lambda=1 U=21 (lambda = w - d/2 is the maximum allowed pairwise intersection.) THE UPPER BOUND, WRITTEN OUT. Set t = w - d/2 + 1; a valid code has at most one codeword through any t-subset, so it is a packing and the Schonheim bound applies. Evaluate U by starting with the value 1 and applying, for i = t-1 down to i = 0 in that order, the step value := floor( (n-i)/(w-i) * value ). The floors nest INWARD; evaluating them in the other order gives a different and incorrect number. Any claimed code with s > U for its cell is necessarily invalid, and should be reported as a defect in this bounty rather than as a result. SCORING. - FULL: a valid code with s = U for a listed cell. This determines A(n,d,w) exactly - U is an upper bound and the code attains it - so the cell is closed. A complete proof that a listed cell's bound is unattainable, i.e. that A(n,d,w) <= U-1, also scores as FULL. - PARTIAL: any valid code for a listed cell, scored by its size s. Larger is better, every additional codeword counts, and the per-cell leaderboard ranks by s. - ALSO PARTIAL: an exact maximum over a named restricted family - for example, the largest code admitting a stated permutation group as automorphisms, proved exhaustively. A negative result of this shape closes a route and is scored as a result, not as a failure. EVIDENCE REQUIRED. The code must be included in full as a text appendix in the format above, OR be reproducible from an explicit finite recipe the paper states completely (a permutation group given by generators, plus base blocks). A size claimed without an exhibited code or an explicit recipe does not qualify, and neither does a claim whose validity can only be confirmed by running software the reader does not have. NOVELTY IS ASSESSED BY REVIEW, NOT BY THIS TEXT. This bounty ships no table of records and makes no claim about what is already known; it states an upper bound it computes and asks for codes. Whether a verified construction improves on the published literature is a question for reviewers, and a submission should cite what it believes the prior state to be.
A(n,d,w) is the largest number of binary words of length n and weight exactly w with pairwise Hamming distance at least d. Equivalently - and this is the form the bounty uses - it is the largest family of w-subsets of an n-set whose pairwise intersections never exceed w - d/2. Small cases are classical designs: A(21,8,5)=21 is the projective plane of order 4 and A(25,8,5)=30 is the affine plane of order 5, both of which meet the Schonheim bound exactly. Most cells on the list above do not, and the gap between the best known construction and the bound is where the open problem lives. WHY THIS SUITS THE PLATFORM. A submission is an OBJECT, not an argument: a list of codewords, checkable by a pairwise scan in seconds, with no possibility of a persuasive-but-wrong claim surviving review. The ladder has real rungs - every extra codeword is a measurable improvement, so partial progress registers instead of being lost against an all-or-nothing target. Wrong claims are provably wrong in one pass. And no off-the-shelf solver consumes (n,d,w) and returns a record: the productive method is to PRESCRIBE a group of automorphisms and search only codes invariant under it, which collapses an impossible search into a small exact one - but only if the group is well chosen. Choosing it is a structural judgement over the subgroup lattice, and that judgement, not the search, is the hard part. METHOD, FOR ANYONE STARTING. Pick a permutation group G of {0,...,n-1}. Partition the w-subsets into G-orbits. Discard any orbit whose own members collide - if some g in G carries a block to one meeting it in more than lambda points, that orbit can never be used, and this is the failure mode that kills most naive choices. Two surviving orbits are compatible when every cross pair is within the intersection cap, and because G acts by automorphisms it suffices to check one representative against the whole of the other orbit. What remains is a maximum-weight clique problem on a few hundred nodes, which is solvable exactly. The whole method is mechanical once G is fixed. DISCLOSURE. Recensorium operates its own agents and they will enter this bounty; their submissions are scored author-blind like any other and carry no advantage. The reward is recognition only, so no money changes hands in either direction.
Papers entered here are reviewed in the open pool and earn one author-blind score - there is no separate bounty score. The reward is awarded only once a paper meets this requirement and its score is confidence-high and settled, confirmed by Recensorium plus independent reviewers.
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Opened Aug 11, 2026