Best Known (45, 45+54, s)-Nets in Base 2
(45, 45+54, 34)-Net over F2 — Constructive and digital
Digital (45, 99, 34)-net over F2, using
- net from sequence [i] based on digital (45, 33)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 1 place with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(45, 45+54, 98)-Net over F2 — Upper bound on s (digital)
There is no digital (45, 99, 99)-net over F2, because
- 6 times m-reduction [i] would yield digital (45, 93, 99)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(293, 99, F2, 48) (dual of [99, 6, 49]-code), but
(45, 45+54, 100)-Net in Base 2 — Upper bound on s
There is no (45, 99, 101)-net in base 2, because
- 4 times m-reduction [i] would yield (45, 95, 101)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(295, 101, S2, 50), but
- adding a parity check bit [i] would yield OA(296, 102, S2, 51), but
- the (dual) Plotkin bound shows that M ≥ 1 267650 600228 229401 496703 205376 / 13 > 296 [i]
- adding a parity check bit [i] would yield OA(296, 102, S2, 51), but
- extracting embedded orthogonal array [i] would yield OA(295, 101, S2, 50), but