Best Known (65−32, 65, s)-Nets in Base 64
(65−32, 65, 513)-Net over F64 — Constructive and digital
Digital (33, 65, 513)-net over F64, using
- t-expansion [i] based on digital (28, 65, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(65−32, 65, 514)-Net in Base 64 — Constructive
(33, 65, 514)-net in base 64, using
- 1 times m-reduction [i] based on (33, 66, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (6, 22, 257)-net in base 64, using
- 2 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- base change [i] based on digital (0, 18, 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, 18, 257)-net over F256, using
- 2 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- (11, 44, 257)-net in base 64, using
- base change [i] based on digital (0, 33, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 33, 257)-net over F256, using
- (6, 22, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(65−32, 65, 1540)-Net over F64 — Digital
Digital (33, 65, 1540)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6465, 1540, F64, 2, 32) (dual of [(1540, 2), 3015, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6465, 2052, F64, 2, 32) (dual of [(2052, 2), 4039, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6465, 4104, F64, 32) (dual of [4104, 4039, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- linear OA(6463, 4096, F64, 32) (dual of [4096, 4033, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(6457, 4096, F64, 29) (dual of [4096, 4039, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(6465, 4104, F64, 32) (dual of [4104, 4039, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(6465, 2052, F64, 2, 32) (dual of [(2052, 2), 4039, 33]-NRT-code), using
(65−32, 65, 2348566)-Net in Base 64 — Upper bound on s
There is no (33, 65, 2348567)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 2521 734774 321405 441396 428338 367761 409149 567523 235722 363996 203829 177781 544031 444928 388221 813313 566331 287609 283803 079189 > 6465 [i]