Best Known (154−79, 154, s)-Nets in Base 2
(154−79, 154, 50)-Net over F2 — Constructive and digital
Digital (75, 154, 50)-net over F2, using
- net from sequence [i] based on digital (75, 49)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
(154−79, 154, 161)-Net over F2 — Upper bound on s (digital)
There is no digital (75, 154, 162)-net over F2, because
- 3 times m-reduction [i] would yield digital (75, 151, 162)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2151, 162, F2, 76) (dual of [162, 11, 77]-code), but
- residual code [i] would yield linear OA(275, 85, F2, 38) (dual of [85, 10, 39]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2151, 162, F2, 76) (dual of [162, 11, 77]-code), but
(154−79, 154, 181)-Net in Base 2 — Upper bound on s
There is no (75, 154, 182)-net in base 2, because
- 1 times m-reduction [i] would yield (75, 153, 182)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 12728 807223 187060 220876 010216 840196 441988 685196 > 2153 [i]