Best Known (40, 75, s)-Nets in Base 64
(40, 75, 513)-Net over F64 — Constructive and digital
Digital (40, 75, 513)-net over F64, using
- t-expansion [i] based on digital (28, 75, 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
(40, 75, 545)-Net in Base 64 — Constructive
(40, 75, 545)-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
- (17, 52, 288)-net in base 64, using
- 4 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 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, 48, 288)-net over F128, using
- 4 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- (6, 23, 257)-net in base 64, using
(40, 75, 2330)-Net over F64 — Digital
Digital (40, 75, 2330)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6475, 2330, F64, 35) (dual of [2330, 2255, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(6475, 4116, F64, 35) (dual of [4116, 4041, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(27) [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(6455, 4096, F64, 28) (dual of [4096, 4041, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(34) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(6475, 4116, F64, 35) (dual of [4116, 4041, 36]-code), using
(40, 75, 8294675)-Net in Base 64 — Upper bound on s
There is no (40, 75, 8294676)-net in base 64, because
- 1 times m-reduction [i] would yield (40, 74, 8294676)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 45 427458 376243 422723 385873 882317 276572 244487 921413 689698 933341 990164 206261 494495 648473 959802 493791 051318 607423 468352 255654 373361 606469 > 6474 [i]