Best Known (68, 68+8, s)-Nets in Base 5
(68, 68+8, 4194306)-Net over F5 — Constructive and digital
Digital (68, 76, 4194306)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (64, 72, 4194300)-net over F5, using
- net defined by OOA [i] based on linear OOA(572, 4194300, F5, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(572, 8388601, F5, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(572, 8388602, F5, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- trace code [i] based on linear OOA(2536, 4194301, F25, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2536, 8388602, F25, 8) (dual of [8388602, 8388566, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- OOA 2-folding [i] based on linear OA(2536, 8388602, F25, 8) (dual of [8388602, 8388566, 9]-code), using
- trace code [i] based on linear OOA(2536, 4194301, F25, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(572, 8388602, F5, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(572, 8388601, F5, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(572, 4194300, F5, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
- digital (0, 4, 6)-net over F5, using
(68, 68+8, large)-Net over F5 — Digital
Digital (68, 76, large)-net over F5, using
- 54 times duplication [i] based on digital (64, 72, large)-net over F5, using
- t-expansion [i] based on digital (63, 72, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(572, large, F5, 9) (dual of [large, large−72, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(571, large, F5, 9) (dual of [large, large−71, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, 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(571, large, F5, 9) (dual of [large, large−71, 10]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(572, large, F5, 9) (dual of [large, large−72, 10]-code), using
- t-expansion [i] based on digital (63, 72, large)-net over F5, using
(68, 68+8, large)-Net in Base 5 — Upper bound on s
There is no (68, 76, large)-net in base 5, because
- 6 times m-reduction [i] would yield (68, 70, large)-net in base 5, but