Best Known (45, 45+40, s)-Nets in Base 64
(45, 45+40, 513)-Net over F64 — Constructive and digital
Digital (45, 85, 513)-net over F64, using
- t-expansion [i] based on digital (28, 85, 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
(45, 45+40, 545)-Net in Base 64 — Constructive
(45, 85, 545)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 27, 257)-net in base 64, using
- 1 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- base change [i] based on digital (0, 21, 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, 21, 257)-net over F256, using
- 1 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- (18, 58, 288)-net in base 64, using
- 5 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- base change [i] based on digital (9, 54, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 54, 288)-net over F128, using
- 5 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- (7, 27, 257)-net in base 64, using
(45, 45+40, 2327)-Net over F64 — Digital
Digital (45, 85, 2327)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6485, 2327, F64, 40) (dual of [2327, 2242, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(6485, 4116, F64, 40) (dual of [4116, 4031, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(32) [i] based on
- linear OA(6479, 4096, F64, 40) (dual of [4096, 4017, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(6465, 4096, F64, 33) (dual of [4096, 4031, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(646, 20, F64, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(39) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(6485, 4116, F64, 40) (dual of [4116, 4031, 41]-code), using
(45, 45+40, 6255037)-Net in Base 64 — Upper bound on s
There is no (45, 85, 6255038)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 3351 961743 825585 369936 313025 407375 132162 136121 111757 034448 340630 633174 263614 636550 257565 431672 483742 398726 423181 119636 869857 242227 893766 855243 133660 573131 > 6485 [i]