Best Known (180−92, 180, s)-Nets in Base 2
(180−92, 180, 52)-Net over F2 — Constructive and digital
Digital (88, 180, 52)-net over F2, using
- t-expansion [i] based on digital (85, 180, 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
(180−92, 180, 57)-Net over F2 — Digital
Digital (88, 180, 57)-net over F2, using
- t-expansion [i] based on digital (83, 180, 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
(180−92, 180, 186)-Net over F2 — Upper bound on s (digital)
There is no digital (88, 180, 187)-net over F2, because
- 4 times m-reduction [i] would yield digital (88, 176, 187)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2176, 187, F2, 88) (dual of [187, 11, 89]-code), but
- residual code [i] would yield linear OA(288, 98, F2, 44) (dual of [98, 10, 45]-code), but
- adding a parity check bit [i] would yield linear OA(289, 99, F2, 45) (dual of [99, 10, 46]-code), but
- residual code [i] would yield linear OA(288, 98, F2, 44) (dual of [98, 10, 45]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2176, 187, F2, 88) (dual of [187, 11, 89]-code), but
(180−92, 180, 209)-Net in Base 2 — Upper bound on s
There is no (88, 180, 210)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 644100 042908 754838 418740 097809 021436 070752 163681 635600 > 2180 [i]