Best Known (57−25, 57, s)-Nets in Base 64
(57−25, 57, 513)-Net over F64 — Constructive and digital
Digital (32, 57, 513)-net over F64, using
- t-expansion [i] based on digital (28, 57, 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
(57−25, 57, 545)-Net in Base 64 — Constructive
(32, 57, 545)-net in base 64, using
- 641 times duplication [i] based on (31, 56, 545)-net in base 64, using
- (u, u+v)-construction [i] based on
- (4, 16, 257)-net in base 64, using
- base change [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- (15, 40, 288)-net in base 64, using
- 2 times m-reduction [i] based on (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
- 2 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
- (4, 16, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(57−25, 57, 3728)-Net over F64 — Digital
Digital (32, 57, 3728)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6457, 3728, F64, 25) (dual of [3728, 3671, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, 4122, F64, 25) (dual of [4122, 4065, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(15) [i] based on
- linear OA(6449, 4096, F64, 25) (dual of [4096, 4047, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(24) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(6457, 4122, F64, 25) (dual of [4122, 4065, 26]-code), using
(57−25, 57, large)-Net in Base 64 — Upper bound on s
There is no (32, 57, large)-net in base 64, because
- 23 times m-reduction [i] would yield (32, 34, large)-net in base 64, but