Best Known (44−24, 44, s)-Nets in Base 64
(44−24, 44, 257)-Net over F64 — Constructive and digital
Digital (20, 44, 257)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (7, 31, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- digital (1, 13, 80)-net over F64, using
(44−24, 44, 322)-Net in Base 64 — Constructive
(20, 44, 322)-net in base 64, using
- (u, u+v)-construction [i] based on
- digital (0, 12, 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
- (8, 32, 257)-net in base 64, using
- base change [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- digital (0, 12, 65)-net over F64, using
(44−24, 44, 459)-Net over F64 — Digital
Digital (20, 44, 459)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6444, 459, F64, 24) (dual of [459, 415, 25]-code), using
- construction XX applied to C1 = C([22,44]), C2 = C([21,43]), C3 = C1 + C2 = C([22,43]), and C∩ = C1 ∩ C2 = C([21,44]) [i] based on
- linear OA(6442, 455, F64, 23) (dual of [455, 413, 24]-code), using the BCH-code C(I) with length 455 | 642−1, defining interval I = {22,23,…,44}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(6442, 455, F64, 23) (dual of [455, 413, 24]-code), using the BCH-code C(I) with length 455 | 642−1, defining interval I = {21,22,…,43}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(6444, 455, F64, 24) (dual of [455, 411, 25]-code), using the BCH-code C(I) with length 455 | 642−1, defining interval I = {21,22,…,44}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(6440, 455, F64, 22) (dual of [455, 415, 23]-code), using the BCH-code C(I) with length 455 | 642−1, defining interval I = {22,23,…,43}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([22,44]), C2 = C([21,43]), C3 = C1 + C2 = C([22,43]), and C∩ = C1 ∩ C2 = C([21,44]) [i] based on
(44−24, 44, 513)-Net in Base 64
(20, 44, 513)-net in base 64, using
- 4 times m-reduction [i] based on (20, 48, 513)-net in base 64, using
- base change [i] based on digital (8, 36, 513)-net over F256, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- K1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F256 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- base change [i] based on digital (8, 36, 513)-net over F256, using
(44−24, 44, 352106)-Net in Base 64 — Upper bound on s
There is no (20, 44, 352107)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 29 643519 286554 975533 006156 973134 837754 543077 988824 838026 072614 807712 910824 132680 > 6444 [i]