Best Known (68−34, 68, s)-Nets in Base 64
(68−34, 68, 513)-Net over F64 — Constructive and digital
Digital (34, 68, 513)-net over F64, using
- t-expansion [i] based on digital (28, 68, 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
(68−34, 68, 514)-Net in Base 64 — Constructive
(34, 68, 514)-net in base 64, using
- base change [i] based on digital (17, 51, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 17, 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
- digital (0, 34, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 17, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(68−34, 68, 1367)-Net over F64 — Digital
Digital (34, 68, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6468, 1367, F64, 3, 34) (dual of [(1367, 3), 4033, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6468, 4101, F64, 34) (dual of [4101, 4033, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- linear OA(6467, 4096, F64, 34) (dual of [4096, 4029, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 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(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- OOA 3-folding [i] based on linear OA(6468, 4101, F64, 34) (dual of [4101, 4033, 35]-code), using
(68−34, 68, 1911272)-Net in Base 64 — Upper bound on s
There is no (34, 68, 1911273)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 661 057277 341878 473879 358200 231060 925774 067606 242400 829544 308140 426402 917066 872531 626823 743131 603628 724578 783565 563895 898928 > 6468 [i]