Best Known (137−66, 137, s)-Nets in Base 2
(137−66, 137, 49)-Net over F2 — Constructive and digital
Digital (71, 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
(137−66, 137, 175)-Net over F2 — Upper bound on s (digital)
There is no digital (71, 137, 176)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2137, 176, F2, 66) (dual of [176, 39, 67]-code), but
- residual code [i] would yield OA(271, 109, S2, 33), but
- 1 times truncation [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]
- 1 times truncation [i] would yield OA(270, 108, S2, 32), but
- residual code [i] would yield OA(271, 109, S2, 33), but
(137−66, 137, 189)-Net in Base 2 — Upper bound on s
There is no (71, 137, 190)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 197029 143559 351823 616045 265994 580115 198041 > 2137 [i]