Best Known (54, 54+8, s)-Nets in Base 8
(54, 54+8, 4194309)-Net over F8 — Constructive and digital
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
(54, 54+8, large)-Net over F8 — Digital
Digital (54, 62, large)-net over F8, using
- 84 times duplication [i] based on digital (50, 58, large)-net over F8, using
- t-expansion [i] based on digital (49, 58, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(858, large, F8, 9) (dual of [large, large−58, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(857, large, F8, 9) (dual of [large, large−57, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times code embedding in larger space [i] based on linear OA(857, large, F8, 9) (dual of [large, large−57, 10]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(858, large, F8, 9) (dual of [large, large−58, 10]-code), using
- t-expansion [i] based on digital (49, 58, large)-net over F8, using
(54, 54+8, large)-Net in Base 8 — Upper bound on s
There is no (54, 62, large)-net in base 8, because
- 6 times m-reduction [i] would yield (54, 56, large)-net in base 8, but