Best Known (72, 72+8, s)-Nets in Base 7
(72, 72+8, 4194400)-Net over F7 — Constructive and digital
Digital (72, 80, 4194400)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (4, 8, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 4, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- trace code for nets [i] based on digital (0, 4, 50)-net over F49, using
- digital (64, 72, 4194300)-net over F7, using
- net defined by OOA [i] based on linear OOA(772, 4194300, F7, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(772, 8388601, F7, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(772, 8388602, F7, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- trace code [i] based on linear OOA(4936, 4194301, F49, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4936, 8388602, F49, 8) (dual of [8388602, 8388566, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(4936, large, F49, 8) (dual of [large, large−36, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 25679568 | 495−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(4936, large, F49, 8) (dual of [large, large−36, 9]-code), using
- OOA 2-folding [i] based on linear OA(4936, 8388602, F49, 8) (dual of [8388602, 8388566, 9]-code), using
- trace code [i] based on linear OOA(4936, 4194301, F49, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(772, 8388602, F7, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(772, 8388601, F7, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(772, 4194300, F7, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
- digital (4, 8, 100)-net over F7, using
(72, 72+8, large)-Net over F7 — Digital
Digital (72, 80, large)-net over F7, using
- t-expansion [i] based on digital (71, 80, large)-net over F7, using
- 2 times m-reduction [i] based on digital (71, 82, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(782, large, F7, 11) (dual of [large, large−82, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(782, large, F7, 11) (dual of [large, large−82, 12]-code), using
- 2 times m-reduction [i] based on digital (71, 82, large)-net over F7, using
(72, 72+8, large)-Net in Base 7 — Upper bound on s
There is no (72, 80, large)-net in base 7, because
- 6 times m-reduction [i] would yield (72, 74, large)-net in base 7, but