Best Known (48, 48+40, s)-Nets in Base 64
(48, 48+40, 578)-Net over F64 — Constructive and digital
Digital (48, 88, 578)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 20, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (28, 68, 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
- digital (0, 20, 65)-net over F64, using
(48, 48+40, 3238)-Net over F64 — Digital
Digital (48, 88, 3238)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6488, 3238, F64, 40) (dual of [3238, 3150, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(6488, 4125, F64, 40) (dual of [4125, 4037, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(29) [i] based on
- 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(6459, 4096, F64, 30) (dual of [4096, 4037, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(649, 29, F64, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,64)), using
- discarding factors / shortening the dual code based on linear OA(649, 64, F64, 9) (dual of [64, 55, 10]-code or 64-arc in PG(8,64)), using
- Reed–Solomon code RS(55,64) [i]
- discarding factors / shortening the dual code based on linear OA(649, 64, F64, 9) (dual of [64, 55, 10]-code or 64-arc in PG(8,64)), using
- construction X applied to Ce(39) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(6488, 4125, F64, 40) (dual of [4125, 4037, 41]-code), using
(48, 48+40, large)-Net in Base 64 — Upper bound on s
There is no (48, 88, large)-net in base 64, because
- 38 times m-reduction [i] would yield (48, 50, large)-net in base 64, but