Best Known (107−55, 107, s)-Nets in Base 2
(107−55, 107, 36)-Net over F2 — Constructive and digital
Digital (52, 107, 36)-net over F2, using
- t-expansion [i] based on digital (51, 107, 36)-net over F2, using
- net from sequence [i] based on digital (51, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (51, 35)-sequence over F2, using
(107−55, 107, 40)-Net over F2 — Digital
Digital (52, 107, 40)-net over F2, using
- t-expansion [i] based on digital (50, 107, 40)-net over F2, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 50 and N(F) ≥ 40, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
(107−55, 107, 114)-Net over F2 — Upper bound on s (digital)
There is no digital (52, 107, 115)-net over F2, because
- 3 times m-reduction [i] would yield digital (52, 104, 115)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2104, 115, F2, 52) (dual of [115, 11, 53]-code), but
- adding a parity check bit [i] would yield linear OA(2105, 116, F2, 53) (dual of [116, 11, 54]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2104, 115, F2, 52) (dual of [115, 11, 53]-code), but
(107−55, 107, 115)-Net in Base 2 — Upper bound on s
There is no (52, 107, 116)-net in base 2, because
- 3 times m-reduction [i] would yield (52, 104, 116)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2104, 116, S2, 52), but
- the linear programming bound shows that M ≥ 2271 629875 608987 087482 092144 033792 / 99 > 2104 [i]
- extracting embedded orthogonal array [i] would yield OA(2104, 116, S2, 52), but