Best Known (89−41, 89, s)-Nets in Base 64
(89−41, 89, 578)-Net over F64 — Constructive and digital
Digital (48, 89, 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, 69, 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
(89−41, 89, 2889)-Net over F64 — Digital
Digital (48, 89, 2889)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6489, 2889, F64, 41) (dual of [2889, 2800, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(6489, 4122, F64, 41) (dual of [4122, 4033, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(31) [i] based on
- linear OA(6481, 4096, F64, 41) (dual of [4096, 4015, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(6463, 4096, F64, 32) (dual of [4096, 4033, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(648, 26, F64, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,64)), using
- discarding factors / shortening the dual code based on linear OA(648, 64, F64, 8) (dual of [64, 56, 9]-code or 64-arc in PG(7,64)), using
- Reed–Solomon code RS(56,64) [i]
- discarding factors / shortening the dual code based on linear OA(648, 64, F64, 8) (dual of [64, 56, 9]-code or 64-arc in PG(7,64)), using
- construction X applied to Ce(40) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(6489, 4122, F64, 41) (dual of [4122, 4033, 42]-code), using
(89−41, 89, large)-Net in Base 64 — Upper bound on s
There is no (48, 89, large)-net in base 64, because
- 39 times m-reduction [i] would yield (48, 50, large)-net in base 64, but