Best Known (71−35, 71, s)-Nets in Base 64
(71−35, 71, 513)-Net over F64 — Constructive and digital
Digital (36, 71, 513)-net over F64, using
- t-expansion [i] based on digital (28, 71, 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
(71−35, 71, 515)-Net in Base 64 — Constructive
(36, 71, 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, 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
- (6, 23, 257)-net in base 64, using
(71−35, 71, 1577)-Net over F64 — Digital
Digital (36, 71, 1577)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6471, 1577, F64, 2, 35) (dual of [(1577, 2), 3083, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6471, 2052, F64, 2, 35) (dual of [(2052, 2), 4033, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6471, 4104, F64, 35) (dual of [4104, 4033, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(31) [i] based on
- linear OA(6469, 4096, F64, 35) (dual of [4096, 4027, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(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(34) ⊂ Ce(31) [i] based on
- OOA 2-folding [i] based on linear OA(6471, 4104, F64, 35) (dual of [4104, 4033, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(6471, 2052, F64, 2, 35) (dual of [(2052, 2), 4033, 36]-NRT-code), using
(71−35, 71, 3117562)-Net in Base 64 — Upper bound on s
There is no (36, 71, 3117563)-net in base 64, because
- 1 times m-reduction [i] would yield (36, 70, 3117563)-net in base 64, but
- 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]