Best Known (45−5, 45, s)-Nets in Base 2
(45−5, 45, 2097153)-Net over F2 — Constructive and digital
Digital (40, 45, 2097153)-net over F2, using
- 21 times duplication [i] based on digital (39, 44, 2097153)-net over F2, using
(45−5, 45, 2097163)-Net in Base 2 — Constructive
(40, 45, 2097163)-net in base 2, using
- net defined by OOA [i] based on OOA(245, 2097163, S2, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(245, 4194327, S2, 5), using
- construction X4 applied to RM(1,22) ⊂ K(22) [i] based on
- OA(244, 4194304, S2, 5), using Kerdock OA K(22) [i]
- linear OA(223, 4194304, F2, 3) (dual of [4194304, 4194281, 4]-code or 4194304-cap in PG(22,2)), using Reed–Muller code RM(1,22) [i]
- linear OA(222, 23, F2, 21) (dual of [23, 1, 22]-code), using
- strength reduction [i] based on linear OA(222, 23, F2, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,2)), using
- dual of repetition code with length 23 [i]
- strength reduction [i] based on linear OA(222, 23, F2, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,2)), using
- linear OA(21, 23, F2, 1) (dual of [23, 22, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to RM(1,22) ⊂ K(22) [i] based on
- OOA 2-folding and stacking with additional row [i] based on OA(245, 4194327, S2, 5), using
(45−5, 45, 5931639)-Net in Base 2 — Upper bound on s
There is no (40, 45, 5931640)-net in base 2, because
- 1 times m-reduction [i] would yield (40, 44, 5931640)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 17 592191 373901 > 244 [i]