Best Known (135−64, 135, s)-Nets in Base 2
(135−64, 135, 49)-Net over F2 — Constructive and digital
Digital (71, 135, 49)-net over F2, using
- t-expansion [i] based on 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]
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
(135−64, 135, 181)-Net over F2 — Upper bound on s (digital)
There is no digital (71, 135, 182)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2135, 182, F2, 64) (dual of [182, 47, 65]-code), but
- residual code [i] would yield OA(271, 117, S2, 32), but
- the linear programming bound shows that M ≥ 436299 229870 540803 344540 378506 426337 722368 / 173 222223 038567 257555 > 271 [i]
- residual code [i] would yield OA(271, 117, S2, 32), but
(135−64, 135, 194)-Net in Base 2 — Upper bound on s
There is no (71, 135, 195)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 46278 039884 203133 343694 668691 141694 169570 > 2135 [i]