Best Known (71−34, 71, s)-Nets in Base 64
(71−34, 71, 513)-Net over F64 — Constructive and digital
Digital (37, 71, 513)-net over F64, using
- t-expansion [i] based on digital (28, 71, 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
(71−34, 71, 516)-Net in Base 64 — Constructive
(37, 71, 516)-net in base 64, using
- 1 times m-reduction [i] based on (37, 72, 516)-net in base 64, using
- base change [i] based on digital (19, 54, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 18, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 36, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 18, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (19, 54, 516)-net over F256, using
(71−34, 71, 2050)-Net over F64 — Digital
Digital (37, 71, 2050)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6471, 2050, F64, 2, 34) (dual of [(2050, 2), 4029, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6471, 2055, F64, 2, 34) (dual of [(2055, 2), 4039, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6471, 4110, F64, 34) (dual of [4110, 4039, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- 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(6457, 4096, F64, 29) (dual of [4096, 4039, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(33) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(6471, 4110, F64, 34) (dual of [4110, 4039, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(6471, 2055, F64, 2, 34) (dual of [(2055, 2), 4039, 35]-NRT-code), using
(71−34, 71, 3981633)-Net in Base 64 — Upper bound on s
There is no (37, 71, 3981634)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 173 292144 756698 199073 068256 605493 296734 472388 357926 343700 473614 144848 180836 819118 040114 033291 684901 966178 031127 172596 848432 713140 > 6471 [i]