Best Known (82−20, 82, s)-Nets in Base 64
(82−20, 82, 838860)-Net over F64 — Constructive and digital
Digital (62, 82, 838860)-net over F64, using
- 641 times duplication [i] based on digital (61, 81, 838860)-net over F64, using
- t-expansion [i] based on 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
- net defined by OOA [i] based on linear OOA(6481, 838860, F64, 21, 21) (dual of [(838860, 21), 17615979, 22]-NRT-code), using
- t-expansion [i] based on digital (60, 81, 838860)-net over F64, using
(82−20, 82, large)-Net over F64 — Digital
Digital (62, 82, large)-net over F64, using
- 642 times duplication [i] based on digital (60, 80, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6480, large, F64, 20) (dual of [large, large−80, 21]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6477, large, F64, 20) (dual of [large, large−77, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 3 times code embedding in larger space [i] based on linear OA(6477, large, F64, 20) (dual of [large, large−77, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6480, large, F64, 20) (dual of [large, large−80, 21]-code), using
(82−20, 82, large)-Net in Base 64 — Upper bound on s
There is no (62, 82, large)-net in base 64, because
- 18 times m-reduction [i] would yield (62, 64, large)-net in base 64, but