Best Known (87−45, 87, s)-Nets in Base 64
(87−45, 87, 513)-Net over F64 — Constructive and digital
Digital (42, 87, 513)-net over F64, using
- t-expansion [i] based on digital (28, 87, 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
(87−45, 87, 1078)-Net over F64 — Digital
Digital (42, 87, 1078)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6487, 1078, F64, 45) (dual of [1078, 991, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(6487, 1366, F64, 45) (dual of [1366, 1279, 46]-code), using
- an extension Ce(44) of the narrow-sense BCH-code C(I) with length 1365 | 642−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- discarding factors / shortening the dual code based on linear OA(6487, 1366, F64, 45) (dual of [1366, 1279, 46]-code), using
(87−45, 87, 1652081)-Net in Base 64 — Upper bound on s
There is no (42, 87, 1652082)-net in base 64, because
- 1 times m-reduction [i] would yield (42, 86, 1652082)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 214527 724932 355531 517285 714809 031706 489201 874554 353030 524527 463161 838078 062517 433798 414612 717117 964972 433094 987950 127859 646584 650922 515862 169279 171032 485680 > 6486 [i]