Best Known (99, 99+97, s)-Nets in Base 2
(99, 99+97, 54)-Net over F2 — Constructive and digital
Digital (99, 196, 54)-net over F2, using
- t-expansion [i] based on digital (95, 196, 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+97, 65)-Net over F2 — Digital
Digital (99, 196, 65)-net over F2, using
- t-expansion [i] based on digital (95, 196, 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+97, 215)-Net over F2 — Upper bound on s (digital)
There is no digital (99, 196, 216)-net over F2, because
- 1 times m-reduction [i] would yield digital (99, 195, 216)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2195, 216, F2, 96) (dual of [216, 21, 97]-code), but
- residual code [i] would yield OA(299, 119, S2, 48), but
- the linear programming bound shows that M ≥ 7 222484 930221 945231 285920 393945 153536 / 10 168125 > 299 [i]
- residual code [i] would yield OA(299, 119, S2, 48), but
- extracting embedded orthogonal array [i] would yield linear OA(2195, 216, F2, 96) (dual of [216, 21, 97]-code), but
(99, 99+97, 248)-Net in Base 2 — Upper bound on s
There is no (99, 196, 249)-net in base 2, because
- 1 times m-reduction [i] would yield (99, 195, 249)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 56211 268133 255866 634800 016350 772202 522650 577072 332620 704933 > 2195 [i]