Best Known (81−20, 81, s)-Nets in Base 81
(81−20, 81, 838860)-Net over F81 — Constructive and digital
Digital (61, 81, 838860)-net over F81, using
- t-expansion [i] based on digital (60, 81, 838860)-net over F81, using
- net defined by OOA [i] based on linear OOA(8181, 838860, F81, 21, 21) (dual of [(838860, 21), 17615979, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8181, 8388601, F81, 21) (dual of [8388601, 8388520, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8181, 8388601, F81, 21) (dual of [8388601, 8388520, 22]-code), using
- net defined by OOA [i] based on linear OOA(8181, 838860, F81, 21, 21) (dual of [(838860, 21), 17615979, 22]-NRT-code), using
(81−20, 81, large)-Net over F81 — Digital
Digital (61, 81, large)-net over F81, using
- t-expansion [i] based on digital (60, 81, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
(81−20, 81, large)-Net in Base 81 — Upper bound on s
There is no (61, 81, large)-net in base 81, because
- 18 times m-reduction [i] would yield (61, 63, large)-net in base 81, but