Best Known (67, 87, s)-Nets in Base 64
(67, 87, 838925)-Net over F64 — Constructive and digital
Digital (67, 87, 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 (57, 77, 838860)-net over F64, using
- net defined by OOA [i] based on linear OOA(6477, 838860, F64, 20, 20) (dual of [(838860, 20), 16777123, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6477, 8388600, F64, 20) (dual of [8388600, 8388523, 21]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(6477, large, F64, 20) (dual of [large, large−77, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(6477, 8388600, F64, 20) (dual of [8388600, 8388523, 21]-code), using
- net defined by OOA [i] based on linear OOA(6477, 838860, F64, 20, 20) (dual of [(838860, 20), 16777123, 21]-NRT-code), using
- digital (0, 10, 65)-net over F64, using
(67, 87, large)-Net over F64 — Digital
Digital (67, 87, large)-net over F64, using
- t-expansion [i] based on digital (66, 87, large)-net over F64, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on digital (66, 88, large)-net over F64, using
(67, 87, large)-Net in Base 64 — Upper bound on s
There is no (67, 87, large)-net in base 64, because
- 18 times m-reduction [i] would yield (67, 69, large)-net in base 64, but