Best Known (72, 72+65, s)-Nets in Base 2
(72, 72+65, 49)-Net over F2 — Constructive and digital
Digital (72, 137, 49)-net over F2, using
- t-expansion [i] based on digital (70, 137, 49)-net over F2, using
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, and 1 place with degree 2 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
(72, 72+65, 190)-Net over F2 — Upper bound on s (digital)
There is no digital (72, 137, 191)-net over F2, because
- 1 times m-reduction [i] would yield digital (72, 136, 191)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2136, 191, F2, 64) (dual of [191, 55, 65]-code), but
- residual code [i] would yield OA(272, 126, S2, 32), but
- the linear programming bound shows that M ≥ 163 005211 721460 055257 870637 096174 404209 999872 / 34370 163308 084940 061297 > 272 [i]
- residual code [i] would yield OA(272, 126, S2, 32), but
- extracting embedded orthogonal array [i] would yield linear OA(2136, 191, F2, 64) (dual of [191, 55, 65]-code), but
(72, 72+65, 199)-Net in Base 2 — Upper bound on s
There is no (72, 137, 200)-net in base 2, because
- 1 times m-reduction [i] would yield (72, 136, 200)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 90738 474423 294661 230022 774832 133413 245074 > 2136 [i]