Best Known (81−39, 81, s)-Nets in Base 64
(81−39, 81, 513)-Net over F64 — Constructive and digital
Digital (42, 81, 513)-net over F64, using
- t-expansion [i] based on digital (28, 81, 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
(81−39, 81, 516)-Net in Base 64 — Constructive
(42, 81, 516)-net in base 64, using
- 641 times duplication [i] based on (41, 80, 516)-net in base 64, using
- base change [i] based on digital (21, 60, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 20, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 40, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 20, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (21, 60, 516)-net over F256, using
(81−39, 81, 2055)-Net over F64 — Digital
Digital (42, 81, 2055)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6481, 2055, F64, 2, 39) (dual of [(2055, 2), 4029, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6481, 4110, F64, 39) (dual of [4110, 4029, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(33) [i] based on
- linear OA(6477, 4096, F64, 39) (dual of [4096, 4019, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(6467, 4096, F64, 34) (dual of [4096, 4029, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(38) ⊂ Ce(33) [i] based on
- OOA 2-folding [i] based on linear OA(6481, 4110, F64, 39) (dual of [4110, 4029, 40]-code), using
(81−39, 81, 5068068)-Net in Base 64 — Upper bound on s
There is no (42, 81, 5068069)-net in base 64, because
- 1 times m-reduction [i] would yield (42, 80, 5068069)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 3 121755 058939 399820 645753 093898 953150 379694 671888 940241 585498 254347 961361 848169 036418 910999 397974 848580 590306 170257 282244 807614 764559 589758 618408 > 6480 [i]