Best Known (81−21, 81, s)-Nets in Base 64
(81−21, 81, 838860)-Net over F64 — Constructive and digital
Digital (60, 81, 838860)-net over F64, using
- net defined by OOA [i] based on linear OOA(6481, 838860, F64, 21, 21) (dual of [(838860, 21), 17615979, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6481, 8388601, F64, 21) (dual of [8388601, 8388520, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6481, large, F64, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, 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(6481, large, F64, 21) (dual of [large, large−81, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6481, 8388601, F64, 21) (dual of [8388601, 8388520, 22]-code), using
(81−21, 81, 5068069)-Net over F64 — Digital
Digital (60, 81, 5068069)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6481, 5068069, F64, 21) (dual of [5068069, 5067988, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6481, large, F64, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, 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(6481, large, F64, 21) (dual of [large, large−81, 22]-code), using
(81−21, 81, large)-Net in Base 64 — Upper bound on s
There is no (60, 81, large)-net in base 64, because
- 19 times m-reduction [i] would yield (60, 62, large)-net in base 64, but