Best Known (52−20, 52, s)-Nets in Base 64
(52−20, 52, 520)-Net over F64 — Constructive and digital
Digital (32, 52, 520)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 65)-net over F64, using
- digital (0, 2, 65)-net over F64 (see above)
- digital (0, 3, 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 (0, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 5, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 6, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 10, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 20, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
(52−20, 52, 6553)-Net in Base 64 — Constructive
(32, 52, 6553)-net in base 64, using
- base change [i] based on digital (19, 39, 6553)-net over F256, using
- net defined by OOA [i] based on linear OOA(25639, 6553, F256, 20, 20) (dual of [(6553, 20), 131021, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(25639, 65530, F256, 20) (dual of [65530, 65491, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(25639, 65530, F256, 20) (dual of [65530, 65491, 21]-code), using
- net defined by OOA [i] based on linear OOA(25639, 6553, F256, 20, 20) (dual of [(6553, 20), 131021, 21]-NRT-code), using
(52−20, 52, 11053)-Net over F64 — Digital
Digital (32, 52, 11053)-net over F64, using
(52−20, 52, 13107)-Net in Base 64
(32, 52, 13107)-net in base 64, using
- base change [i] based on digital (19, 39, 13107)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25639, 13107, F256, 5, 20) (dual of [(13107, 5), 65496, 21]-NRT-code), using
- OOA 5-folding [i] based on linear OA(25639, 65535, F256, 20) (dual of [65535, 65496, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using
- OOA 5-folding [i] based on linear OA(25639, 65535, F256, 20) (dual of [65535, 65496, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25639, 13107, F256, 5, 20) (dual of [(13107, 5), 65496, 21]-NRT-code), using
(52−20, 52, large)-Net in Base 64 — Upper bound on s
There is no (32, 52, large)-net in base 64, because
- 18 times m-reduction [i] would yield (32, 34, large)-net in base 64, but