Best Known (57−30, 57, s)-Nets in Base 64
(57−30, 57, 305)-Net over F64 — Constructive and digital
Digital (27, 57, 305)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 20, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- digital (7, 37, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- digital (5, 20, 128)-net over F64, using
(57−30, 57, 354)-Net in Base 64 — Constructive
(27, 57, 354)-net in base 64, using
- (u, u+v)-construction [i] based on
- (5, 20, 257)-net in base 64, using
- base change [i] based on digital (0, 15, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 15, 257)-net over F256, using
- digital (7, 37, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- (5, 20, 257)-net in base 64, using
(57−30, 57, 722)-Net over F64 — Digital
Digital (27, 57, 722)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6457, 722, F64, 30) (dual of [722, 665, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, 819, F64, 30) (dual of [819, 762, 31]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 819 | 642−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(6457, 819, F64, 30) (dual of [819, 762, 31]-code), using
(57−30, 57, 744570)-Net in Base 64 — Upper bound on s
There is no (27, 57, 744571)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 8 959086 168924 072379 492352 076307 512093 938272 050891 737638 009027 080751 971096 965277 771862 678443 809191 541776 > 6457 [i]