Best Known (22, 44, s)-Nets in Base 64
(22, 44, 372)-Net over F64 — Constructive and digital
Digital (22, 44, 372)-net over F64, using
- 1 times m-reduction [i] based on digital (22, 45, 372)-net over F64, using
- net defined by OOA [i] based on linear OOA(6445, 372, F64, 23, 23) (dual of [(372, 23), 8511, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6445, 4093, F64, 23) (dual of [4093, 4048, 24]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6445, 4093, F64, 23) (dual of [4093, 4048, 24]-code), using
- net defined by OOA [i] based on linear OOA(6445, 372, F64, 23, 23) (dual of [(372, 23), 8511, 24]-NRT-code), using
(22, 44, 514)-Net in Base 64 — Constructive
(22, 44, 514)-net in base 64, using
- base change [i] based on digital (11, 33, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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
- digital (0, 22, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 11, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(22, 44, 1367)-Net over F64 — Digital
Digital (22, 44, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6444, 1367, F64, 3, 22) (dual of [(1367, 3), 4057, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6444, 4101, F64, 22) (dual of [4101, 4057, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(6443, 4096, F64, 22) (dual of [4096, 4053, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- OOA 3-folding [i] based on linear OA(6444, 4101, F64, 22) (dual of [4101, 4057, 23]-code), using
(22, 44, 1307349)-Net in Base 64 — Upper bound on s
There is no (22, 44, 1307350)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 29 642840 559255 572161 251307 819048 339031 088544 125940 831446 745630 670533 073154 016036 > 6444 [i]