Best Known (51, 51+206, s)-Nets in Base 2
(51, 51+206, 36)-Net over F2 — Constructive and digital
Digital (51, 257, 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]
(51, 51+206, 40)-Net over F2 — Digital
Digital (51, 257, 40)-net over F2, using
- t-expansion [i] based on digital (50, 257, 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
(51, 51+206, 62)-Net in Base 2 — Upper bound on s
There is no (51, 257, 63)-net in base 2, because
- 13 times m-reduction [i] would yield (51, 244, 63)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2244, 63, S2, 4, 193), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 3844 659212 957764 301173 255361 616590 690440 604959 601346 852872 615094 012740 632576 / 97 > 2244 [i]
- extracting embedded OOA [i] would yield OOA(2244, 63, S2, 4, 193), but