Best Known (170−84, 170, s)-Nets in Base 2
(170−84, 170, 52)-Net over F2 — Constructive and digital
Digital (86, 170, 52)-net over F2, using
- t-expansion [i] based on digital (85, 170, 52)-net over F2, using
- net from sequence [i] based on digital (85, 51)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 3 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (85, 51)-sequence over F2, using
(170−84, 170, 57)-Net over F2 — Digital
Digital (86, 170, 57)-net over F2, using
- t-expansion [i] based on digital (83, 170, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
(170−84, 170, 187)-Net over F2 — Upper bound on s (digital)
There is no digital (86, 170, 188)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2170, 188, F2, 84) (dual of [188, 18, 85]-code), but
- residual code [i] would yield OA(286, 103, S2, 42), but
- the linear programming bound shows that M ≥ 158456 325028 528675 187087 900672 / 1705 > 286 [i]
- residual code [i] would yield OA(286, 103, S2, 42), but
(170−84, 170, 216)-Net in Base 2 — Upper bound on s
There is no (86, 170, 217)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1649 074240 012269 249945 823406 035232 990951 929036 470200 > 2170 [i]