Best Known (29−5, 29, s)-Nets in Base 2
(29−5, 29, 8193)-Net over F2 — Constructive and digital
Digital (24, 29, 8193)-net over F2, using
- 21 times duplication [i] based on digital (23, 28, 8193)-net over F2, using
(29−5, 29, 8199)-Net in Base 2 — Constructive
(24, 29, 8199)-net in base 2, using
- net defined by OOA [i] based on OOA(229, 8199, S2, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(229, 16399, S2, 5), using
- construction X4 applied to RM(1,14) ⊂ K(14) [i] based on
- OA(228, 16384, S2, 5), using Kerdock OA K(14) [i]
- linear OA(215, 16384, F2, 3) (dual of [16384, 16369, 4]-code or 16384-cap in PG(14,2)), using Reed–Muller code RM(1,14) [i]
- linear OA(214, 15, F2, 13) (dual of [15, 1, 14]-code), using
- strength reduction [i] based on linear OA(214, 15, F2, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
- dual of repetition code with length 15 [i]
- strength reduction [i] based on linear OA(214, 15, F2, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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(1,14) ⊂ K(14) [i] based on
- OOA 2-folding and stacking with additional row [i] based on OA(229, 16399, S2, 5), using
(29−5, 29, 23167)-Net in Base 2 — Upper bound on s
There is no (24, 29, 23168)-net in base 2, because
- 1 times m-reduction [i] would yield (24, 28, 23168)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 268 436033 > 228 [i]