Best Known (156−86, 156, s)-Nets in Base 2
(156−86, 156, 49)-Net over F2 — Constructive and digital
Digital (70, 156, 49)-net over F2, using
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, and 1 place with degree 2 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
(156−86, 156, 149)-Net over F2 — Upper bound on s (digital)
There is no digital (70, 156, 150)-net over F2, because
- 14 times m-reduction [i] would yield digital (70, 142, 150)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2142, 150, F2, 72) (dual of [150, 8, 73]-code), but
- residual code [i] would yield linear OA(270, 77, F2, 36) (dual of [77, 7, 37]-code), but
- residual code [i] would yield linear OA(234, 40, F2, 18) (dual of [40, 6, 19]-code), but
- residual code [i] would yield linear OA(270, 77, F2, 36) (dual of [77, 7, 37]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2142, 150, F2, 72) (dual of [150, 8, 73]-code), but
(156−86, 156, 151)-Net in Base 2 — Upper bound on s
There is no (70, 156, 152)-net in base 2, because
- 10 times m-reduction [i] would yield (70, 146, 152)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2146, 152, S2, 76), but
- adding a parity check bit [i] would yield OA(2147, 153, S2, 77), but
- the (dual) Plotkin bound shows that M ≥ 2854 495385 411919 762116 571938 898990 272765 493248 / 13 > 2147 [i]
- adding a parity check bit [i] would yield OA(2147, 153, S2, 77), but
- extracting embedded orthogonal array [i] would yield OA(2146, 152, S2, 76), but