Best Known (67, 67+81, s)-Nets in Base 2
(67, 67+81, 43)-Net over F2 — Constructive and digital
Digital (67, 148, 43)-net over F2, using
- t-expansion [i] based on digital (59, 148, 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
(67, 67+81, 48)-Net over F2 — Digital
Digital (67, 148, 48)-net over F2, using
- t-expansion [i] based on digital (65, 148, 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
(67, 67+81, 143)-Net over F2 — Upper bound on s (digital)
There is no digital (67, 148, 144)-net over F2, because
- 13 times m-reduction [i] would yield digital (67, 135, 144)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2135, 144, F2, 68) (dual of [144, 9, 69]-code), but
- residual code [i] would yield linear OA(267, 75, F2, 34) (dual of [75, 8, 35]-code), but
- adding a parity check bit [i] would yield linear OA(268, 76, F2, 35) (dual of [76, 8, 36]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(268, 76, F2, 35) (dual of [76, 8, 36]-code), but
- residual code [i] would yield linear OA(267, 75, F2, 34) (dual of [75, 8, 35]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2135, 144, F2, 68) (dual of [144, 9, 69]-code), but
(67, 67+81, 145)-Net in Base 2 — Upper bound on s
There is no (67, 148, 146)-net in base 2, because
- 13 times m-reduction [i] would yield (67, 135, 146)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2135, 146, S2, 68), but
- the linear programming bound shows that M ≥ 9 756576 024357 147624 421876 744283 658158 866432 / 205 > 2135 [i]
- extracting embedded orthogonal array [i] would yield OA(2135, 146, S2, 68), but