Best Known (47, 47+39, s)-Nets in Base 64
(47, 47+39, 578)-Net over F64 — Constructive and digital
Digital (47, 86, 578)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 19, 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, 67, 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, 19, 65)-net over F64, using
(47, 47+39, 3263)-Net over F64 — Digital
Digital (47, 86, 3263)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6486, 3263, F64, 39) (dual of [3263, 3177, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(6486, 4126, F64, 39) (dual of [4126, 4040, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,14]) [i] based on
- linear OA(6477, 4097, F64, 39) (dual of [4097, 4020, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(6457, 4097, F64, 29) (dual of [4097, 4040, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [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 C([0,19]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6486, 4126, F64, 39) (dual of [4126, 4040, 40]-code), using
(47, 47+39, large)-Net in Base 64 — Upper bound on s
There is no (47, 86, large)-net in base 64, because
- 37 times m-reduction [i] would yield (47, 49, large)-net in base 64, but