Best Known (79−8, 79, s)-Nets in Base 5
(79−8, 79, 4194344)-Net over F5 — Constructive and digital
Digital (71, 79, 4194344)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 44)-net 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
(79−8, 79, large)-Net over F5 — Digital
Digital (71, 79, large)-net over F5, using
- 2 times m-reduction [i] based on digital (71, 81, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(581, large, F5, 10) (dual of [large, large−81, 11]-code), using
- strength reduction [i] based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- strength reduction [i] based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(581, large, F5, 10) (dual of [large, large−81, 11]-code), using
(79−8, 79, large)-Net in Base 5 — Upper bound on s
There is no (71, 79, large)-net in base 5, because
- 6 times m-reduction [i] would yield (71, 73, large)-net in base 5, but