Best Known (159−77, 159, s)-Nets in Base 2
(159−77, 159, 51)-Net over F2 — Constructive and digital
Digital (82, 159, 51)-net over F2, using
- t-expansion [i] based on digital (80, 159, 51)-net over F2, using
- net from sequence [i] based on digital (80, 50)-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 2 places 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]
- net from sequence [i] based on digital (80, 50)-sequence over F2, using
(159−77, 159, 56)-Net over F2 — Digital
Digital (82, 159, 56)-net over F2, using
- t-expansion [i] based on digital (80, 159, 56)-net over F2, using
- net from sequence [i] based on digital (80, 55)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 80 and N(F) ≥ 56, using
- net from sequence [i] based on digital (80, 55)-sequence over F2, using
(159−77, 159, 189)-Net over F2 — Upper bound on s (digital)
There is no digital (82, 159, 190)-net over F2, because
- 3 times m-reduction [i] would yield digital (82, 156, 190)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2156, 190, F2, 74) (dual of [190, 34, 75]-code), but
- construction Y1 [i] would yield
- linear OA(2155, 178, F2, 74) (dual of [178, 23, 75]-code), but
- adding a parity check bit [i] would yield linear OA(2156, 179, F2, 75) (dual of [179, 23, 76]-code), but
- OA(234, 190, S2, 12), but
- discarding factors would yield OA(234, 154, S2, 12), but
- the Rao or (dual) Hamming bound shows that M ≥ 17486 314616 > 234 [i]
- discarding factors would yield OA(234, 154, S2, 12), but
- linear OA(2155, 178, F2, 74) (dual of [178, 23, 75]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2156, 190, F2, 74) (dual of [190, 34, 75]-code), but
(159−77, 159, 216)-Net in Base 2 — Upper bound on s
There is no (82, 159, 217)-net in base 2, because
- 1 times m-reduction [i] would yield (82, 158, 217)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 388102 958415 665324 150660 051198 683953 226955 148200 > 2158 [i]