Best Known (114−57, 114, s)-Nets in Base 2
(114−57, 114, 42)-Net over F2 — Constructive and digital
Digital (57, 114, 42)-net over F2, using
- t-expansion [i] based on digital (54, 114, 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
(114−57, 114, 126)-Net over F2 — Upper bound on s (digital)
There is no digital (57, 114, 127)-net over F2, because
- 1 times m-reduction [i] would yield digital (57, 113, 127)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2113, 127, F2, 56) (dual of [127, 14, 57]-code), but
- residual code [i] would yield linear OA(257, 70, F2, 28) (dual of [70, 13, 29]-code), but
- adding a parity check bit [i] would yield linear OA(258, 71, F2, 29) (dual of [71, 13, 30]-code), but
- residual code [i] would yield linear OA(257, 70, F2, 28) (dual of [70, 13, 29]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2113, 127, F2, 56) (dual of [127, 14, 57]-code), but
(114−57, 114, 127)-Net in Base 2 — Upper bound on s
There is no (57, 114, 128)-net in base 2, because
- 1 times m-reduction [i] would yield (57, 113, 128)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2113, 128, S2, 56), but
- the linear programming bound shows that M ≥ 5948 959895 135390 989180 988498 284682 149888 / 569415 > 2113 [i]
- extracting embedded orthogonal array [i] would yield OA(2113, 128, S2, 56), but