Best Known (53−8, 53, s)-Nets in Base 8
(53−8, 53, 524296)-Net over F8 — Constructive and digital
Digital (45, 53, 524296)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (41, 49, 524287)-net over F8, using
- net defined by OOA [i] based on linear OOA(849, 524287, F8, 8, 8) (dual of [(524287, 8), 4194247, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(849, 2097148, F8, 8) (dual of [2097148, 2097099, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(849, 2097151, F8, 8) (dual of [2097151, 2097102, 9]-code), using
- the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(849, 2097151, F8, 8) (dual of [2097151, 2097102, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(849, 2097148, F8, 8) (dual of [2097148, 2097099, 9]-code), using
- net defined by OOA [i] based on linear OOA(849, 524287, F8, 8, 8) (dual of [(524287, 8), 4194247, 9]-NRT-code), using
- digital (0, 4, 9)-net over F8, using
(53−8, 53, 1048577)-Net in Base 8 — Constructive
(45, 53, 1048577)-net in base 8, using
- 81 times duplication [i] based on (44, 52, 1048577)-net in base 8, using
- net defined by OOA [i] based on OOA(852, 1048577, S8, 8, 8), using
- OA 4-folding and stacking [i] based on OA(852, 4194308, S8, 8), using
- discarding factors based on OA(852, 4194310, S8, 8), using
- trace code [i] based on OA(6426, 2097155, S64, 8), using
- discarding parts of the base [i] based on linear OA(12822, 2097155, F128, 8) (dual of [2097155, 2097133, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding parts of the base [i] based on linear OA(12822, 2097155, F128, 8) (dual of [2097155, 2097133, 9]-code), using
- trace code [i] based on OA(6426, 2097155, S64, 8), using
- discarding factors based on OA(852, 4194310, S8, 8), using
- OA 4-folding and stacking [i] based on OA(852, 4194308, S8, 8), using
- net defined by OOA [i] based on OOA(852, 1048577, S8, 8, 8), using
(53−8, 53, 3322786)-Net over F8 — Digital
Digital (45, 53, 3322786)-net over F8, using
(53−8, 53, large)-Net in Base 8 — Upper bound on s
There is no (45, 53, large)-net in base 8, because
- 6 times m-reduction [i] would yield (45, 47, large)-net in base 8, but