Best Known (3, 27, s)-Nets in Base 8
(3, 27, 24)-Net over F8 — Constructive and digital
Digital (3, 27, 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
(3, 27, 33)-Net over F8 — Upper bound on s (digital)
There is no digital (3, 27, 34)-net over F8, because
- 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
(3, 27, 56)-Net in Base 8 — Upper bound on s
There is no (3, 27, 57)-net in base 8, because
- extracting embedded orthogonal array [i] would yield OA(827, 57, S8, 24), but
- the linear programming bound shows that M ≥ 1 534162 066694 529730 322075 289108 086784 / 622080 959743 > 827 [i]