Best Known (87, 172, s)-Nets in Base 2
(87, 172, 52)-Net over F2 — Constructive and digital
Digital (87, 172, 52)-net over F2, using
- t-expansion [i] based on digital (85, 172, 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
(87, 172, 57)-Net over F2 — Digital
Digital (87, 172, 57)-net over F2, using
- t-expansion [i] based on digital (83, 172, 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
(87, 172, 191)-Net over F2 — Upper bound on s (digital)
There is no digital (87, 172, 192)-net over F2, because
- 1 times m-reduction [i] would yield digital (87, 171, 192)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2171, 192, F2, 84) (dual of [192, 21, 85]-code), but
- residual code [i] would yield OA(287, 107, S2, 42), but
- the linear programming bound shows that M ≥ 1155 305065 783002 570789 057883 799552 / 7 380945 > 287 [i]
- residual code [i] would yield OA(287, 107, S2, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(2171, 192, F2, 84) (dual of [192, 21, 85]-code), but
(87, 172, 220)-Net in Base 2 — Upper bound on s
There is no (87, 172, 221)-net in base 2, because
- 1 times m-reduction [i] would yield (87, 171, 221)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3068 350693 082351 777703 878332 103193 786329 642918 201740 > 2171 [i]