Best Known (215−163, 215, s)-Nets in Base 2
(215−163, 215, 36)-Net over F2 — Constructive and digital
Digital (52, 215, 36)-net over F2, using
- t-expansion [i] based on digital (51, 215, 36)-net over F2, using
- net from sequence [i] based on digital (51, 35)-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 3 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]
- net from sequence [i] based on digital (51, 35)-sequence over F2, using
(215−163, 215, 40)-Net over F2 — Digital
Digital (52, 215, 40)-net over F2, using
- t-expansion [i] based on digital (50, 215, 40)-net over F2, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 50 and N(F) ≥ 40, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
(215−163, 215, 67)-Net in Base 2 — Upper bound on s
There is no (52, 215, 68)-net in base 2, because
- 18 times m-reduction [i] would yield (52, 197, 68)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2197, 68, S2, 3, 145), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 19 283256 531107 883306 503545 108093 951230 266435 925393 514023 616512 / 73 > 2197 [i]
- extracting embedded OOA [i] would yield OOA(2197, 68, S2, 3, 145), but