Best Known (40, 40+33, s)-Nets in Base 64
(40, 40+33, 513)-Net over F64 — Constructive and digital
Digital (40, 73, 513)-net over F64, using
- t-expansion [i] based on digital (28, 73, 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
(40, 40+33, 547)-Net in Base 64 — Constructive
(40, 73, 547)-net in base 64, using
- (u, u+v)-construction [i] based on
- (8, 24, 259)-net in base 64, using
- base change [i] based on digital (2, 18, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- base change [i] based on digital (2, 18, 259)-net over F256, using
- (16, 49, 288)-net in base 64, using
- base change [i] based on digital (9, 42, 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, 42, 288)-net over F128, using
- (8, 24, 259)-net in base 64, using
(40, 40+33, 3073)-Net over F64 — Digital
Digital (40, 73, 3073)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6473, 3073, F64, 33) (dual of [3073, 3000, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(6473, 4122, F64, 33) (dual of [4122, 4049, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(23) [i] based on
- linear OA(6465, 4096, F64, 33) (dual of [4096, 4031, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(32) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(6473, 4122, F64, 33) (dual of [4122, 4049, 34]-code), using
(40, 40+33, large)-Net in Base 64 — Upper bound on s
There is no (40, 73, large)-net in base 64, because
- 31 times m-reduction [i] would yield (40, 42, large)-net in base 64, but