Almost difference sets at sixteen undetermined cells of Gordon's table, and the status of (61,30,14,30)

Abstract

A (v,k,λ,t)-almost difference set is a k-element subset D of a group of order v in which t of the v-1 nonzero elements have exactly λ representations as a difference of two elements of D, and the remaining v-1-t have exactly λ+1. Gordon's table of cyclic almost difference sets for 4 ≤ v ≤ 63 marks 94 of its cells as undetermined: at those parameters the exhaustive search reported there did not run to completion. We exhibit almost difference sets at sixteen of them, namely at (57,27), (58,28), (58,29), (60,29), (61,22), (61,28), (62,19), (62,23), (62,27), (62,28), (62,29), (63,19), (63,21), (63,25), (63,26) and (63,27), each by an explicit witness printed in full, and at a seventeenth cell, (63,24), which turns out to be an instance of a product construction of Cai and Ding. Three of the sixteen are specializations of published constructions to cells the table leaves undetermined; for the other thirteen the recorded literature search matched the parameters to no published family, with the gaps in that search stated explicitly. We also record that the cell (61,30,14,30), named in the source as the first open case of an existence question of Arasu, Ding, Helleseth, Kumar and Martinsen, is realized by the quadratic residues modulo 61, and prove that for every odd modulus the question splits on v mod 4: at v ≡ 3 (mod 4) every almost difference set with those parameters is a difference set, so nothing but difference-set existence survives there, while at v ≡ 1 (mod 4) exactly one parameter tuple is admissible — the Paley shape, which the quadratic residues realize at every prime. Every theorem, proposition and corollary is machine-checked in Lean 4. No nonexistence is claimed anywhere.

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