Best Known (19, 19+18, s)-Nets in Base 64
(19, 19+18, 456)-Net over F64 — Constructive and digital
Digital (19, 37, 456)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 456, F64, 18, 18) (dual of [(456, 18), 8171, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6437, 4104, F64, 18) (dual of [4104, 4067, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- OA 9-folding and stacking [i] based on linear OA(6437, 4104, F64, 18) (dual of [4104, 4067, 19]-code), using
(19, 19+18, 514)-Net in Base 64 — Constructive
(19, 37, 514)-net in base 64, using
- 1 times m-reduction [i] based on (19, 38, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (3, 12, 257)-net in base 64, using
- base change [i] based on digital (0, 9, 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, 9, 257)-net over F256, using
- (7, 26, 257)-net in base 64, using
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, 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 (see above)
- base change [i] based on digital (0, 21, 257)-net over F256, using
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- (3, 12, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(19, 19+18, 1664)-Net over F64 — Digital
Digital (19, 37, 1664)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6437, 1664, F64, 2, 18) (dual of [(1664, 2), 3291, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6437, 2052, F64, 2, 18) (dual of [(2052, 2), 4067, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6437, 4104, F64, 18) (dual of [4104, 4067, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(6437, 4104, F64, 18) (dual of [4104, 4067, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(6437, 2052, F64, 2, 18) (dual of [(2052, 2), 4067, 19]-NRT-code), using
(19, 19+18, 1753139)-Net in Base 64 — Upper bound on s
There is no (19, 37, 1753140)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 6 740009 955614 830917 637718 330647 523771 813720 736141 389727 624598 522210 > 6437 [i]