Best Known (84−40, 84, s)-Nets in Base 64
(84−40, 84, 513)-Net over F64 — Constructive and digital
Digital (44, 84, 513)-net over F64, using
- t-expansion [i] based on digital (28, 84, 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
(84−40, 84, 545)-Net in Base 64 — Constructive
(44, 84, 545)-net in base 64, using
- base change [i] based on (32, 72, 545)-net in base 128, using
- (u, u+v)-construction [i] based on
- (3, 23, 257)-net in base 128, using
- 1 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
- base change [i] based on digital (0, 21, 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, 21, 257)-net over F256, using
- 1 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
- digital (9, 49, 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
- (3, 23, 257)-net in base 128, using
- (u, u+v)-construction [i] based on
(84−40, 84, 2084)-Net over F64 — Digital
Digital (44, 84, 2084)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6484, 2084, F64, 40) (dual of [2084, 2000, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(6484, 4113, F64, 40) (dual of [4113, 4029, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- linear OA(6479, 4096, F64, 40) (dual of [4096, 4017, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(6467, 4096, F64, 34) (dual of [4096, 4029, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(6484, 4113, F64, 40) (dual of [4113, 4029, 41]-code), using
(84−40, 84, 5080667)-Net in Base 64 — Upper bound on s
There is no (44, 84, 5080668)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 52 374451 399275 314160 292715 101339 713917 161266 928798 997649 122271 912860 012068 713582 407790 815124 507736 256096 802629 921886 983414 186950 057810 574813 659050 918934 > 6484 [i]