Best Known (4, 4+26, s)-Nets in Base 8
(4, 4+26, 25)-Net over F8 — Constructive and digital
Digital (4, 30, 25)-net over F8, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 4 and N(F) ≥ 25, using
(4, 4+26, 75)-Net over F8 — Upper bound on s (digital)
There is no digital (4, 30, 76)-net over F8, because
- 2 times m-reduction [i] would yield digital (4, 28, 76)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(828, 76, F8, 24) (dual of [76, 48, 25]-code), but
- construction Y1 [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
- linear OA(848, 76, F8, 42) (dual of [76, 28, 43]-code), but
- discarding factors / shortening the dual code would yield linear OA(848, 58, F8, 42) (dual of [58, 10, 43]-code), but
- construction Y1 [i] would yield
- linear OA(847, 50, F8, 42) (dual of [50, 3, 43]-code), but
- “Hi4†bound on codes from Brouwer’s database [i]
- OA(810, 58, S8, 8), but
- discarding factors would yield OA(810, 57, S8, 8), but
- the linear programming bound shows that M ≥ 1426 626655 027200 / 1 315507 > 810 [i]
- discarding factors would yield OA(810, 57, S8, 8), but
- linear OA(847, 50, F8, 42) (dual of [50, 3, 43]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(848, 58, F8, 42) (dual of [58, 10, 43]-code), but
- linear OA(827, 34, F8, 24) (dual of [34, 7, 25]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(828, 76, F8, 24) (dual of [76, 48, 25]-code), but
(4, 4+26, 79)-Net in Base 8 — Upper bound on s
There is no (4, 30, 80)-net in base 8, because
- extracting embedded orthogonal array [i] would yield OA(830, 80, S8, 26), but
- the linear programming bound shows that M ≥ 13 682212 274302 658089 590997 718068 536897 550187 233280 / 10656 691984 486719 330503 > 830 [i]