Best Known (135−65, 135, s)-Nets in Base 2
(135−65, 135, 49)-Net over F2 — Constructive and digital
Digital (70, 135, 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]
(135−65, 135, 172)-Net over F2 — Upper bound on s (digital)
There is no digital (70, 135, 173)-net over F2, because
- 1 times m-reduction [i] would yield digital (70, 134, 173)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2134, 173, F2, 64) (dual of [173, 39, 65]-code), but
- residual code [i] would yield OA(270, 108, S2, 32), but
- the linear programming bound shows that M ≥ 4 297708 243581 570312 211812 843520 / 3462 071301 > 270 [i]
- residual code [i] would yield OA(270, 108, S2, 32), but
- extracting embedded orthogonal array [i] would yield linear OA(2134, 173, F2, 64) (dual of [173, 39, 65]-code), but
(135−65, 135, 189)-Net in Base 2 — Upper bound on s
There is no (70, 135, 190)-net in base 2, because
- 1 times m-reduction [i] would yield (70, 134, 190)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 23253 983258 442103 145218 112637 005428 967271 > 2134 [i]