Best Known (75, 75+68, s)-Nets in Base 2
(75, 75+68, 50)-Net over F2 — Constructive and digital
Digital (75, 143, 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]
(75, 75+68, 185)-Net over F2 — Upper bound on s (digital)
There is no digital (75, 143, 186)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2143, 186, F2, 68) (dual of [186, 43, 69]-code), but
- residual code [i] would yield OA(275, 117, S2, 34), but
- the linear programming bound shows that M ≥ 19 511733 253817 452295 327962 524563 800064 / 514 656942 650235 > 275 [i]
- residual code [i] would yield OA(275, 117, S2, 34), but
(75, 75+68, 203)-Net in Base 2 — Upper bound on s
There is no (75, 143, 204)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 11 959968 508277 990836 375274 533751 715141 083190 > 2143 [i]