Best Known (66, 66+83, s)-Nets in Base 2
(66, 66+83, 43)-Net over F2 — Constructive and digital
Digital (66, 149, 43)-net over F2, using
- t-expansion [i] based on digital (59, 149, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
(66, 66+83, 48)-Net over F2 — Digital
Digital (66, 149, 48)-net over F2, using
- t-expansion [i] based on digital (65, 149, 48)-net over F2, using
- net from sequence [i] based on digital (65, 47)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 65 and N(F) ≥ 48, using
- net from sequence [i] based on digital (65, 47)-sequence over F2, using
(66, 66+83, 141)-Net over F2 — Upper bound on s (digital)
There is no digital (66, 149, 142)-net over F2, because
- 15 times m-reduction [i] would yield digital (66, 134, 142)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2134, 142, F2, 68) (dual of [142, 8, 69]-code), but
- residual code [i] would yield linear OA(266, 73, F2, 34) (dual of [73, 7, 35]-code), but
- “Hel†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(266, 73, F2, 34) (dual of [73, 7, 35]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2134, 142, F2, 68) (dual of [142, 8, 69]-code), but
(66, 66+83, 143)-Net in Base 2 — Upper bound on s
There is no (66, 149, 144)-net in base 2, because
- 1 times m-reduction [i] would yield (66, 148, 144)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 401 291778 987212 339528 976798 240148 928082 062660 > 2148 [i]