Best Known (68−36, 68, s)-Nets in Base 2
(68−36, 68, 21)-Net over F2 — Constructive and digital
Digital (32, 68, 21)-net over F2, using
- t-expansion [i] based on digital (21, 68, 21)-net over F2, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 21 and N(F) ≥ 21, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
(68−36, 68, 27)-Net over F2 — Digital
Digital (32, 68, 27)-net over F2, using
- t-expansion [i] based on digital (31, 68, 27)-net over F2, using
- net from sequence [i] based on digital (31, 26)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 31 and N(F) ≥ 27, using
- net from sequence [i] based on digital (31, 26)-sequence over F2, using
(68−36, 68, 71)-Net over F2 — Upper bound on s (digital)
There is no digital (32, 68, 72)-net over F2, because
- 4 times m-reduction [i] would yield digital (32, 64, 72)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(264, 72, F2, 32) (dual of [72, 8, 33]-code), but
- adding a parity check bit [i] would yield linear OA(265, 73, F2, 33) (dual of [73, 8, 34]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(265, 73, F2, 33) (dual of [73, 8, 34]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(264, 72, F2, 32) (dual of [72, 8, 33]-code), but
(68−36, 68, 75)-Net in Base 2 — Upper bound on s
There is no (32, 68, 76)-net in base 2, because
- 2 times m-reduction [i] would yield (32, 66, 76)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(266, 76, S2, 34), but
- the linear programming bound shows that M ≥ 2361 183241 434822 606848 / 23 > 266 [i]
- extracting embedded orthogonal array [i] would yield OA(266, 76, S2, 34), but