Best Known (49−24, 49, s)-Nets in Base 64
(49−24, 49, 342)-Net over F64 — Constructive and digital
Digital (25, 49, 342)-net over F64, using
- net defined by OOA [i] based on linear OOA(6449, 342, F64, 24, 24) (dual of [(342, 24), 8159, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(6449, 4104, F64, 24) (dual of [4104, 4055, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(6441, 4096, F64, 21) (dual of [4096, 4055, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(23) ⊂ Ce(20) [i] based on
- OA 12-folding and stacking [i] based on linear OA(6449, 4104, F64, 24) (dual of [4104, 4055, 25]-code), using
(49−24, 49, 514)-Net in Base 64 — Constructive
(25, 49, 514)-net in base 64, using
- 1 times m-reduction [i] based on (25, 50, 514)-net in base 64, using
- (u, u+v)-construction [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
- (9, 34, 257)-net in base 64, using
- 2 times m-reduction [i] based on (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
- 2 times m-reduction [i] based on (9, 36, 257)-net in base 64, using
- (4, 16, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(49−24, 49, 1509)-Net over F64 — Digital
Digital (25, 49, 1509)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6449, 1509, F64, 2, 24) (dual of [(1509, 2), 2969, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6449, 2052, F64, 2, 24) (dual of [(2052, 2), 4055, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6449, 4104, F64, 24) (dual of [4104, 4055, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(6441, 4096, F64, 21) (dual of [4096, 4055, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(23) ⊂ Ce(20) [i] based on
- OOA 2-folding [i] based on linear OA(6449, 4104, F64, 24) (dual of [4104, 4055, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(6449, 2052, F64, 2, 24) (dual of [(2052, 2), 4055, 25]-NRT-code), using
(49−24, 49, 1991840)-Net in Base 64 — Upper bound on s
There is no (25, 49, 1991841)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 31828 819379 395580 591126 790828 215201 673640 861075 622051 133339 802436 363418 491490 035863 811560 > 6449 [i]