Best Known (53−26, 53, s)-Nets in Base 64
(53−26, 53, 354)-Net over F64 — Constructive and digital
Digital (27, 53, 354)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 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 (7, 33, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64 (see above)
- digital (7, 20, 177)-net over F64, using
(53−26, 53, 514)-Net in Base 64 — Constructive
(27, 53, 514)-net in base 64, using
- 1 times m-reduction [i] based on (27, 54, 514)-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
- (9, 36, 257)-net in base 64, using
- base change [i] based on digital (0, 27, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 27, 257)-net over F256, using
- (5, 18, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(53−26, 53, 1503)-Net over F64 — Digital
Digital (27, 53, 1503)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6453, 1503, F64, 2, 26) (dual of [(1503, 2), 2953, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6453, 2052, F64, 2, 26) (dual of [(2052, 2), 4051, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6453, 4104, F64, 26) (dual of [4104, 4051, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(6451, 4096, F64, 26) (dual of [4096, 4045, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(6453, 4104, F64, 26) (dual of [4104, 4051, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(6453, 2052, F64, 2, 26) (dual of [(2052, 2), 4051, 27]-NRT-code), using
(53−26, 53, 2078361)-Net in Base 64 — Upper bound on s
There is no (27, 53, 2078362)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 533997 469596 310360 708138 509578 584176 816529 759223 512921 429973 681861 190855 899185 105480 274570 612404 > 6453 [i]