Best Known (127−68, 127, s)-Nets in Base 2
(127−68, 127, 43)-Net over F2 — Constructive and digital
Digital (59, 127, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
(127−68, 127, 127)-Net over F2 — Upper bound on s (digital)
There is no digital (59, 127, 128)-net over F2, because
- 4 times m-reduction [i] would yield digital (59, 123, 128)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2123, 128, F2, 64) (dual of [128, 5, 65]-code), but
(127−68, 127, 129)-Net in Base 2 — Upper bound on s
There is no (59, 127, 130)-net in base 2, because
- 8 times m-reduction [i] would yield (59, 119, 130)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2119, 130, S2, 60), but
- the linear programming bound shows that M ≥ 16610 033035 328308 747805 973025 263185 821696 / 21793 > 2119 [i]
- extracting embedded orthogonal array [i] would yield OA(2119, 130, S2, 60), but