Best Known (69−16, 69, s)-Nets in Base 64
(69−16, 69, 1048640)-Net over F64 — Constructive and digital
Digital (53, 69, 1048640)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 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 (45, 61, 1048575)-net over F64, using
- net defined by OOA [i] based on linear OOA(6461, 1048575, F64, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(6461, 8388600, F64, 16) (dual of [8388600, 8388539, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(6461, 8388600, F64, 16) (dual of [8388600, 8388539, 17]-code), using
- net defined by OOA [i] based on linear OOA(6461, 1048575, F64, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- digital (0, 8, 65)-net over F64, using
(69−16, 69, large)-Net over F64 — Digital
Digital (53, 69, large)-net over F64, using
- 2 times m-reduction [i] based on digital (53, 71, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6471, large, F64, 18) (dual of [large, large−71, 19]-code), using
- 2 times code embedding in larger space [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 2 times code embedding in larger space [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6471, large, F64, 18) (dual of [large, large−71, 19]-code), using
(69−16, 69, large)-Net in Base 64 — Upper bound on s
There is no (53, 69, large)-net in base 64, because
- 14 times m-reduction [i] would yield (53, 55, large)-net in base 64, but