Best Known (45−22, 45, s)-Nets in Base 64
(45−22, 45, 373)-Net over F64 — Constructive and digital
Digital (23, 45, 373)-net over F64, using
- net defined by OOA [i] based on linear OOA(6445, 373, F64, 22, 22) (dual of [(373, 22), 8161, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(6445, 4103, F64, 22) (dual of [4103, 4058, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(6445, 4104, F64, 22) (dual of [4104, 4059, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [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(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(6445, 4104, F64, 22) (dual of [4104, 4059, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(6445, 4103, F64, 22) (dual of [4103, 4058, 23]-code), using
(45−22, 45, 514)-Net in Base 64 — Constructive
(23, 45, 514)-net in base 64, using
- 1 times m-reduction [i] based on (23, 46, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (4, 15, 257)-net in base 64, using
- 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
- base change [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
- (8, 31, 257)-net in base 64, using
- 1 times m-reduction [i] based on (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 (see above)
- base change [i] based on digital (0, 24, 257)-net over F256, using
- 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- (4, 15, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(45−22, 45, 1531)-Net over F64 — Digital
Digital (23, 45, 1531)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6445, 1531, F64, 2, 22) (dual of [(1531, 2), 3017, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6445, 2052, F64, 2, 22) (dual of [(2052, 2), 4059, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6445, 4104, F64, 22) (dual of [4104, 4059, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [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(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(6445, 4104, F64, 22) (dual of [4104, 4059, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(6445, 2052, F64, 2, 22) (dual of [(2052, 2), 4059, 23]-NRT-code), using
(45−22, 45, 1908053)-Net in Base 64 — Upper bound on s
There is no (23, 45, 1908054)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 1897 145144 872807 081463 657394 644365 502392 601726 027034 008463 594931 032013 209145 268548 > 6445 [i]