Best Known (34−18, 34, s)-Nets in Base 2
(34−18, 34, 17)-Net over F2 — Constructive and digital
Digital (16, 34, 17)-net over F2, using
- t-expansion [i] based on digital (15, 34, 17)-net over F2, using
- net from sequence [i] based on digital (15, 16)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 15 and N(F) ≥ 17, using
- net from sequence [i] based on digital (15, 16)-sequence over F2, using
(34−18, 34, 39)-Net over F2 — Upper bound on s (digital)
There is no digital (16, 34, 40)-net over F2, because
- 2 times m-reduction [i] would yield digital (16, 32, 40)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(232, 40, F2, 16) (dual of [40, 8, 17]-code), but
- adding a parity check bit [i] would yield linear OA(233, 41, F2, 17) (dual of [41, 8, 18]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(233, 41, F2, 17) (dual of [41, 8, 18]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(232, 40, F2, 16) (dual of [40, 8, 17]-code), but
(34−18, 34, 41)-Net in Base 2 — Upper bound on s
There is no (16, 34, 42)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(234, 42, S2, 18), but
- the linear programming bound shows that M ≥ 8 005819 039744 / 455 > 234 [i]