Best Known (222−172, 222, s)-Nets in Base 2
(222−172, 222, 35)-Net over F2 — Constructive and digital
Digital (50, 222, 35)-net over F2, using
- t-expansion [i] based on digital (48, 222, 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]
- net from sequence [i] based on digital (48, 34)-sequence over F2, using
(222−172, 222, 40)-Net over F2 — Digital
Digital (50, 222, 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
(222−172, 222, 65)-Net in Base 2 — Upper bound on s
There is no (50, 222, 66)-net in base 2, because
- 33 times m-reduction [i] would yield (50, 189, 66)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2189, 66, S2, 3, 139), but
- the LP bound with quadratic polynomials shows that M ≥ 29816 233243 086733 628219 999760 236415 476486 188361 204163 936256 / 35 > 2189 [i]
- extracting embedded OOA [i] would yield OOA(2189, 66, S2, 3, 139), but