Best Known (55−29, 55, s)-Nets in Base 64
(55−29, 55, 305)-Net over F64 — Constructive and digital
Digital (26, 55, 305)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 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, 36, 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, 19, 128)-net over F64, using
(55−29, 55, 354)-Net in Base 64 — Constructive
(26, 55, 354)-net in base 64, using
- (u, u+v)-construction [i] based on
- (5, 19, 257)-net in base 64, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
- digital (7, 36, 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, 19, 257)-net in base 64, using
(55−29, 55, 698)-Net over F64 — Digital
Digital (26, 55, 698)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6455, 698, F64, 29) (dual of [698, 643, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(6455, 819, F64, 29) (dual of [819, 764, 30]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 819 | 642−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(6455, 819, F64, 29) (dual of [819, 764, 30]-code), using
(55−29, 55, 888808)-Net in Base 64 — Upper bound on s
There is no (26, 55, 888809)-net in base 64, because
- 1 times m-reduction [i] would yield (26, 54, 888809)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 34 176181 327643 392890 996939 812009 575680 103580 102869 790055 454334 982892 447367 397400 941577 352525 344720 > 6454 [i]