Best Known (69−36, 69, s)-Nets in Base 2
(69−36, 69, 24)-Net over F2 — Constructive and digital
Digital (33, 69, 24)-net over F2, using
- net from sequence [i] based on digital (33, 23)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 33 and N(F) ≥ 24, using
(69−36, 69, 28)-Net over F2 — Digital
Digital (33, 69, 28)-net over F2, using
- net from sequence [i] based on digital (33, 27)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 33 and N(F) ≥ 28, using
(69−36, 69, 74)-Net over F2 — Upper bound on s (digital)
There is no digital (33, 69, 75)-net over F2, because
- 2 times m-reduction [i] would yield digital (33, 67, 75)-net over F2, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(267, 75, F2, 34) (dual of [75, 8, 35]-code), but
(69−36, 69, 77)-Net in Base 2 — Upper bound on s
There is no (33, 69, 78)-net in base 2, because
- 2 times m-reduction [i] would yield (33, 67, 78)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(267, 78, S2, 34), but
- the linear programming bound shows that M ≥ 144032 177727 524179 017728 / 759 > 267 [i]
- extracting embedded orthogonal array [i] would yield OA(267, 78, S2, 34), but