Best Known (83−9, 83, s)-Nets in Base 5
(83−9, 83, 4194300)-Net over F5 — Constructive and digital
Digital (74, 83, 4194300)-net over F5, using
- 51 times duplication [i] based on digital (73, 82, 4194300)-net over F5, using
- net defined by OOA [i] based on linear OOA(582, 4194300, F5, 10, 9) (dual of [(4194300, 10), 41942918, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(582, 8388601, F5, 2, 9) (dual of [(8388601, 2), 16777120, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(582, 8388602, F5, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
- trace code [i] based on linear OOA(2541, 4194301, F25, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2541, 8388602, F25, 9) (dual of [8388602, 8388561, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2541, large, F25, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2541, large, F25, 9) (dual of [large, large−41, 10]-code), using
- OOA 2-folding [i] based on linear OA(2541, 8388602, F25, 9) (dual of [8388602, 8388561, 10]-code), using
- trace code [i] based on linear OOA(2541, 4194301, F25, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(582, 8388602, F5, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(582, 8388601, F5, 2, 9) (dual of [(8388601, 2), 16777120, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(582, 4194300, F5, 10, 9) (dual of [(4194300, 10), 41942918, 10]-NRT-code), using
(83−9, 83, large)-Net over F5 — Digital
Digital (74, 83, large)-net over F5, using
- 52 times duplication [i] based on digital (72, 81, large)-net over F5, using
- t-expansion [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
- t-expansion [i] based on digital (71, 81, large)-net over F5, using
(83−9, 83, large)-Net in Base 5 — Upper bound on s
There is no (74, 83, large)-net in base 5, because
- 7 times m-reduction [i] would yield (74, 76, large)-net in base 5, but