Best Known (148−73, 148, s)-Nets in Base 2
(148−73, 148, 50)-Net over F2 — Constructive and digital
Digital (75, 148, 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]
(148−73, 148, 168)-Net over F2 — Upper bound on s (digital)
There is no digital (75, 148, 169)-net over F2, because
- 1 times m-reduction [i] would yield digital (75, 147, 169)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2147, 169, F2, 72) (dual of [169, 22, 73]-code), but
- residual code [i] would yield OA(275, 96, S2, 36), but
- the linear programming bound shows that M ≥ 22708 462595 641194 417680 809984 / 528333 > 275 [i]
- residual code [i] would yield OA(275, 96, S2, 36), but
- extracting embedded orthogonal array [i] would yield linear OA(2147, 169, F2, 72) (dual of [169, 22, 73]-code), but
(148−73, 148, 193)-Net in Base 2 — Upper bound on s
There is no (75, 148, 194)-net in base 2, because
- 1 times m-reduction [i] would yield (75, 147, 194)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 195 413370 295278 872049 903477 268816 810454 782800 > 2147 [i]