Best Known (146−68, 146, s)-Nets in Base 2
(146−68, 146, 50)-Net over F2 — Constructive and digital
Digital (78, 146, 50)-net over F2, using
- t-expansion [i] based on digital (75, 146, 50)-net over F2, using
- net from sequence [i] based on digital (75, 49)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 1 place with degree 6 [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 (75, 49)-sequence over F2, using
(146−68, 146, 52)-Net over F2 — Digital
Digital (78, 146, 52)-net over F2, using
- t-expansion [i] based on digital (77, 146, 52)-net over F2, using
- net from sequence [i] based on digital (77, 51)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 77 and N(F) ≥ 52, using
- net from sequence [i] based on digital (77, 51)-sequence over F2, using
(146−68, 146, 213)-Net over F2 — Upper bound on s (digital)
There is no digital (78, 146, 214)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2146, 214, F2, 68) (dual of [214, 68, 69]-code), but
- residual code [i] would yield OA(278, 145, S2, 34), but
- the linear programming bound shows that M ≥ 6 452658 897421 925882 991923 762933 214377 859980 440698 591468 415644 336128 / 20 712793 240469 221474 764436 259197 787149 263939 > 278 [i]
- residual code [i] would yield OA(278, 145, S2, 34), but
(146−68, 146, 219)-Net in Base 2 — Upper bound on s
There is no (78, 146, 220)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 101 284446 530241 554413 535768 528294 421971 923330 > 2146 [i]