Best Known (99, 99+90, s)-Nets in Base 2
(99, 99+90, 54)-Net over F2 — Constructive and digital
Digital (99, 189, 54)-net over F2, using
- t-expansion [i] based on digital (95, 189, 54)-net over F2, using
- net from sequence [i] based on digital (95, 53)-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 5 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 (95, 53)-sequence over F2, using
(99, 99+90, 65)-Net over F2 — Digital
Digital (99, 189, 65)-net over F2, using
- t-expansion [i] based on digital (95, 189, 65)-net over F2, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
- net from sequence [i] based on digital (95, 64)-sequence over F2, using
(99, 99+90, 238)-Net over F2 — Upper bound on s (digital)
There is no digital (99, 189, 239)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2189, 239, F2, 90) (dual of [239, 50, 91]-code), but
- residual code [i] would yield OA(299, 148, S2, 45), but
- 1 times truncation [i] would yield OA(298, 147, S2, 44), but
- the linear programming bound shows that M ≥ 386 271048 917458 443577 540946 323578 953396 125696 / 1194 549431 128209 > 298 [i]
- 1 times truncation [i] would yield OA(298, 147, S2, 44), but
- residual code [i] would yield OA(299, 148, S2, 45), but
(99, 99+90, 262)-Net in Base 2 — Upper bound on s
There is no (99, 189, 263)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 822 616674 061834 532576 288971 464014 397716 147335 065252 288128 > 2189 [i]