Best Known (90−44, 90, s)-Nets in Base 64
(90−44, 90, 513)-Net over F64 — Constructive and digital
Digital (46, 90, 513)-net over F64, using
- t-expansion [i] based on digital (28, 90, 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
(90−44, 90, 515)-Net in Base 64 — Constructive
(46, 90, 515)-net in base 64, using
- (u, u+v)-construction [i] based on
- (8, 30, 257)-net in base 64, using
- 2 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- base change [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- 2 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- (16, 60, 258)-net in base 64, using
- base change [i] based on digital (1, 45, 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, 45, 258)-net over F256, using
- (8, 30, 257)-net in base 64, using
(90−44, 90, 1909)-Net over F64 — Digital
Digital (46, 90, 1909)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6490, 1909, F64, 2, 44) (dual of [(1909, 2), 3728, 45]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6490, 2053, F64, 2, 44) (dual of [(2053, 2), 4016, 45]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6490, 4106, F64, 44) (dual of [4106, 4016, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(6490, 4107, F64, 44) (dual of [4107, 4017, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(39) [i] based on
- linear OA(6487, 4096, F64, 44) (dual of [4096, 4009, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- 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(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(43) ⊂ Ce(39) [i] based on
- discarding factors / shortening the dual code based on linear OA(6490, 4107, F64, 44) (dual of [4107, 4017, 45]-code), using
- OOA 2-folding [i] based on linear OA(6490, 4106, F64, 44) (dual of [4106, 4016, 45]-code), using
- discarding factors / shortening the dual code based on linear OOA(6490, 2053, F64, 2, 44) (dual of [(2053, 2), 4016, 45]-NRT-code), using
(90−44, 90, 3519080)-Net in Base 64 — Upper bound on s
There is no (46, 90, 3519081)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 3 599140 027082 701157 407562 583348 795524 838256 773164 625638 629632 968549 110278 653147 483190 701797 370226 572187 634077 682941 725699 645490 616734 807514 223154 902191 677359 165944 > 6490 [i]