Best Known (81, 81+6, s)-Nets in Base 8
(81, 81+6, large)-Net over F8 — Constructive and digital
Digital (81, 87, large)-net over F8, using
- t-expansion [i] based on digital (79, 87, large)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (21, 25, 4194301)-net over F8, using
- net defined by OOA [i] based on linear OOA(825, 4194301, F8, 4, 4) (dual of [(4194301, 4), 16777179, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(825, 4194301, F8, 3, 4) (dual of [(4194301, 3), 12582878, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(825, 8388602, F8, 4) (dual of [8388602, 8388577, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(825, large, F8, 4) (dual of [large, large−25, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(825, large, F8, 4) (dual of [large, large−25, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(825, 8388602, F8, 4) (dual of [8388602, 8388577, 5]-code), using
- appending kth column [i] based on linear OOA(825, 4194301, F8, 3, 4) (dual of [(4194301, 3), 12582878, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(825, 4194301, F8, 4, 4) (dual of [(4194301, 4), 16777179, 5]-NRT-code), using
- digital (54, 62, 4194309)-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 (50, 58, 4194300)-net over F8, using
- net defined by OOA [i] based on linear OOA(858, 4194300, F8, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(858, 8388601, F8, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(858, 8388602, F8, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- trace code [i] based on linear OOA(6429, 4194301, F64, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6429, 8388602, F64, 8) (dual of [8388602, 8388573, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- OOA 2-folding [i] based on linear OA(6429, 8388602, F64, 8) (dual of [8388602, 8388573, 9]-code), using
- trace code [i] based on linear OOA(6429, 4194301, F64, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(858, 8388602, F8, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(858, 8388601, F8, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(858, 4194300, F8, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- digital (0, 4, 9)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (21, 25, 4194301)-net over F8, using
- (u, u+v)-construction [i] based on
(81, 81+6, large)-Net in Base 8 — Upper bound on s
There is no (81, 87, large)-net in base 8, because
- 4 times m-reduction [i] would yield (81, 83, large)-net in base 8, but