Best Known (70−34, 70, s)-Nets in Base 64
(70−34, 70, 513)-Net over F64 — Constructive and digital
Digital (36, 70, 513)-net over F64, using
- t-expansion [i] based on digital (28, 70, 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
(70−34, 70, 515)-Net in Base 64 — Constructive
(36, 70, 515)-net in base 64, using
- (u, u+v)-construction [i] based on
- (6, 23, 257)-net in base 64, using
- 1 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
- 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- (13, 47, 258)-net in base 64, using
- 1 times m-reduction [i] based on (13, 48, 258)-net in base 64, using
- base change [i] based on digital (1, 36, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 36, 258)-net over F256, using
- 1 times m-reduction [i] based on (13, 48, 258)-net in base 64, using
- (6, 23, 257)-net in base 64, using
(70−34, 70, 1791)-Net over F64 — Digital
Digital (36, 70, 1791)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6470, 1791, F64, 2, 34) (dual of [(1791, 2), 3512, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6470, 2053, F64, 2, 34) (dual of [(2053, 2), 4036, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6470, 4106, F64, 34) (dual of [4106, 4036, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(6470, 4107, F64, 34) (dual of [4107, 4037, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(29) [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(6459, 4096, F64, 30) (dual of [4096, 4037, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to Ce(33) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(6470, 4107, F64, 34) (dual of [4107, 4037, 35]-code), using
- OOA 2-folding [i] based on linear OA(6470, 4106, F64, 34) (dual of [4106, 4036, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(6470, 2053, F64, 2, 34) (dual of [(2053, 2), 4036, 35]-NRT-code), using
(70−34, 70, 3117562)-Net in Base 64 — Upper bound on s
There is no (36, 70, 3117563)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 2 707693 264730 251512 934041 239129 895531 230021 065667 680390 250968 440708 022318 166816 505926 465187 226391 177500 586513 351703 615683 187550 > 6470 [i]