Best Known (210−162, 210, s)-Nets in Base 2
(210−162, 210, 35)-Net over F2 — Constructive and digital
Digital (48, 210, 35)-net over F2, using
- net from sequence [i] based on digital (48, 34)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(210−162, 210, 36)-Net over F2 — Digital
Digital (48, 210, 36)-net over F2, using
- t-expansion [i] based on digital (47, 210, 36)-net over F2, using
- net from sequence [i] based on digital (47, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 47 and N(F) ≥ 36, using
- net from sequence [i] based on digital (47, 35)-sequence over F2, using
(210−162, 210, 62)-Net in Base 2 — Upper bound on s
There is no (48, 210, 63)-net in base 2, because
- 28 times m-reduction [i] would yield (48, 182, 63)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2182, 63, S2, 3, 134), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 98 079714 615416 886934 934209 737619 787751 599303 819750 539264 / 15 > 2182 [i]
- extracting embedded OOA [i] would yield OOA(2182, 63, S2, 3, 134), but