Best Known (102−99, 102, s)-Nets in Base 8
(102−99, 102, 24)-Net over F8 — Constructive and digital
Digital (3, 102, 24)-net over F8, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- the Klein quartic over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
(102−99, 102, 33)-Net over F8 — Upper bound on s (digital)
There is no digital (3, 102, 34)-net over F8, because
- 75 times m-reduction [i] would yield digital (3, 27, 34)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(827, 34, F8, 24) (dual of [34, 7, 25]-code), but
- construction Y1 [i] would yield
- OA(826, 28, S8, 24), but
- the (dual) Plotkin bound shows that M ≥ 9 671406 556917 033397 649408 / 25 > 826 [i]
- OA(87, 34, S8, 6), but
- discarding factors would yield OA(87, 32, S8, 6), but
- the linear programming bound shows that M ≥ 3784 900608 / 1729 > 87 [i]
- discarding factors would yield OA(87, 32, S8, 6), but
- OA(826, 28, S8, 24), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(827, 34, F8, 24) (dual of [34, 7, 25]-code), but
(102−99, 102, 35)-Net in Base 8 — Upper bound on s
There is no (3, 102, 36)-net in base 8, because
- 33 times m-reduction [i] would yield (3, 69, 36)-net in base 8, but
- extracting embedded OOA [i] would yield OOA(869, 36, S8, 2, 66), but
- the bound derived from the LP bound by Trinker shows that N ≤ 1144 / 3 < 83 [i]
- extracting embedded OOA [i] would yield OOA(869, 36, S8, 2, 66), but