Existence conjecture for D-optimal difference families over abelian groups

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Let (v;r,s;λ)(v;r,s;\lambda) be a D-optimal parameter set, meaning that it is the parameter set of a D-optimal difference family and satisfies the D-optimality conditions. A difference family over a group of order vv is a pair of subsets of that group whose parameters are (v;r,s;λ)(v;r,s;\lambda).

Existence conjecture. For each D-optimal parameter set (v;r,s;λ)(v;r,s;\lambda) there exists a difference family over some abelian group of order vv having these parameters.

D-optimal difference families yield D-optimal matrices, so the conjecture would establish existence of a suitable matrix for every D-optimal parameter set. The source notes that some parameter sets, including (85;39,34;31)(85;39,34;31) and (99;43,42;36)(99;43,42;36), were still unresolved; no resolution of the conjecture is supplied here.

References

Primary source

Dragomir Ž. Djoković, “A few new orders for D-optimal matrices”, arXiv:2210.03931 (2022).

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