Best Known (78, ∞, s)-Nets in Base 2
(78, ∞, 50)-Net over F2 — Constructive and digital
Digital (78, m, 50)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (78, 49)-sequence over F2, using
- t-expansion [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]
- t-expansion [i] based on digital (75, 49)-sequence over F2, using
(78, ∞, 52)-Net over F2 — Digital
Digital (78, m, 52)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (78, 51)-sequence over F2, using
- t-expansion [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
- t-expansion [i] based on digital (77, 51)-sequence over F2, using
(78, ∞, 87)-Net in Base 2 — Upper bound on s
There is no (78, m, 88)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (78, 607, 88)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2607, 88, S2, 7, 529), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 178462 365586 433745 159701 637401 629458 782743 287556 282726 195013 230435 335596 113246 353560 358600 761117 039984 746814 707378 824341 156450 657451 309632 441598 712865 573350 253127 307000 692121 594660 651008 / 265 > 2607 [i]
- extracting embedded OOA [i] would yield OOA(2607, 88, S2, 7, 529), but