Best Known (75, 150, s)-Nets in Base 2
(75, 150, 50)-Net over F2 — Constructive and digital
Digital (75, 150, 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]
(75, 150, 167)-Net over F2 — Upper bound on s (digital)
There is no digital (75, 150, 168)-net over F2, because
- 1 times m-reduction [i] would yield digital (75, 149, 168)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2149, 168, F2, 74) (dual of [168, 19, 75]-code), but
- residual code [i] would yield linear OA(275, 93, F2, 37) (dual of [93, 18, 38]-code), but
- “Bro†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(275, 93, F2, 37) (dual of [93, 18, 38]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2149, 168, F2, 74) (dual of [168, 19, 75]-code), but
(75, 150, 189)-Net in Base 2 — Upper bound on s
There is no (75, 150, 190)-net in base 2, because
- 1 times m-reduction [i] would yield (75, 149, 190)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 832 639342 615682 044423 755393 845235 169379 998167 > 2149 [i]