Best Known (109, 109+90, s)-Nets in Base 2
(109, 109+90, 56)-Net over F2 — Constructive and digital
Digital (109, 199, 56)-net over F2, using
- t-expansion [i] based on digital (105, 199, 56)-net over F2, using
- net from sequence [i] based on digital (105, 55)-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 7 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 (105, 55)-sequence over F2, using
(109, 109+90, 69)-Net over F2 — Digital
Digital (109, 199, 69)-net over F2, using
(109, 109+90, 289)-Net in Base 2 — Upper bound on s
There is no (109, 199, 290)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(2199, 290, S2, 90), but
- 9 times code embedding in larger space [i] would yield OA(2208, 299, S2, 90), but
- adding a parity check bit [i] would yield OA(2209, 300, S2, 91), but
- the linear programming bound shows that M ≥ 437987 836379 661851 428112 482969 460104 960459 291237 979833 080590 515802 667419 590325 215760 903409 709975 863296 / 433 753365 263741 276278 643266 060465 678125 > 2209 [i]
- adding a parity check bit [i] would yield OA(2209, 300, S2, 91), but
- 9 times code embedding in larger space [i] would yield OA(2208, 299, S2, 90), but