Best Known (70, 70+21, s)-Nets in Base 64
(70, 70+21, 838925)-Net over F64 — Constructive and digital
Digital (70, 91, 838925)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- 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
- digital (0, 10, 65)-net over F64, using
(70, 70+21, large)-Net over F64 — Digital
Digital (70, 91, large)-net over F64, using
- 643 times duplication [i] based on digital (67, 88, large)-net over F64, using
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
(70, 70+21, large)-Net in Base 64 — Upper bound on s
There is no (70, 91, large)-net in base 64, because
- 19 times m-reduction [i] would yield (70, 72, large)-net in base 64, but