Best Known (257−189, 257, s)-Nets in Base 2
(257−189, 257, 43)-Net over F2 — Constructive and digital
Digital (68, 257, 43)-net over F2, using
- t-expansion [i] based on digital (59, 257, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
(257−189, 257, 49)-Net over F2 — Digital
Digital (68, 257, 49)-net over F2, using
- net from sequence [i] based on digital (68, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 68 and N(F) ≥ 49, using
(257−189, 257, 86)-Net in Base 2 — Upper bound on s
There is no (68, 257, 87)-net in base 2, because
- 4 times m-reduction [i] would yield (68, 253, 87)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2253, 87, S2, 3, 185), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 694752 535423 897172 541425 910052 127447 119619 907993 843384 236745 504047 478777 839616 / 31 > 2253 [i]
- extracting embedded OOA [i] would yield OOA(2253, 87, S2, 3, 185), but