Best Known (125−69, 125, s)-Nets in Base 2
(125−69, 125, 42)-Net over F2 — Constructive and digital
Digital (56, 125, 42)-net over F2, using
- t-expansion [i] based on digital (54, 125, 42)-net over F2, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
(125−69, 125, 122)-Net over F2 — Upper bound on s (digital)
There is no digital (56, 125, 123)-net over F2, because
- 13 times m-reduction [i] would yield digital (56, 112, 123)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2112, 123, F2, 56) (dual of [123, 11, 57]-code), but
- residual code [i] would yield linear OA(256, 66, F2, 28) (dual of [66, 10, 29]-code), but
- adding a parity check bit [i] would yield linear OA(257, 67, F2, 29) (dual of [67, 10, 30]-code), but
- residual code [i] would yield linear OA(256, 66, F2, 28) (dual of [66, 10, 29]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2112, 123, F2, 56) (dual of [123, 11, 57]-code), but
(125−69, 125, 123)-Net in Base 2 — Upper bound on s
There is no (56, 125, 124)-net in base 2, because
- 13 times m-reduction [i] would yield (56, 112, 124)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2112, 124, S2, 56), but
- the linear programming bound shows that M ≥ 21 267647 932558 653966 460912 964485 513216 / 3451 > 2112 [i]
- extracting embedded orthogonal array [i] would yield OA(2112, 124, S2, 56), but