Best Known (60−27, 60, s)-Nets in Base 64
(60−27, 60, 513)-Net over F64 — Constructive and digital
Digital (33, 60, 513)-net over F64, using
- t-expansion [i] based on digital (28, 60, 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
(60−27, 60, 545)-Net in Base 64 — Constructive
(33, 60, 545)-net in base 64, using
- (u, u+v)-construction [i] based on
- (5, 18, 257)-net in base 64, using
- 2 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
- base change [i] based on digital (0, 15, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 15, 257)-net over F256, using
- 2 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
- (15, 42, 288)-net in base 64, using
- base change [i] based on digital (9, 36, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 36, 288)-net over F128, using
- (5, 18, 257)-net in base 64, using
(60−27, 60, 2945)-Net over F64 — Digital
Digital (33, 60, 2945)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6460, 2945, F64, 27) (dual of [2945, 2885, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6460, 4120, F64, 27) (dual of [4120, 4060, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- linear OA(6453, 4097, F64, 27) (dual of [4097, 4044, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(647, 23, F64, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,64)), using
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- Reed–Solomon code RS(57,64) [i]
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6460, 4120, F64, 27) (dual of [4120, 4060, 28]-code), using
(60−27, 60, large)-Net in Base 64 — Upper bound on s
There is no (33, 60, large)-net in base 64, because
- 25 times m-reduction [i] would yield (33, 35, large)-net in base 64, but