Best Known (44−20, 44, s)-Nets in Base 64
(44−20, 44, 411)-Net over F64 — Constructive and digital
Digital (24, 44, 411)-net over F64, using
- 641 times duplication [i] based on digital (23, 43, 411)-net over F64, using
- net defined by OOA [i] based on linear OOA(6443, 411, F64, 20, 20) (dual of [(411, 20), 8177, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6443, 4110, F64, 20) (dual of [4110, 4067, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- 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(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(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- OA 10-folding and stacking [i] based on linear OA(6443, 4110, F64, 20) (dual of [4110, 4067, 21]-code), using
- net defined by OOA [i] based on linear OOA(6443, 411, F64, 20, 20) (dual of [(411, 20), 8177, 21]-NRT-code), using
(44−20, 44, 517)-Net in Base 64 — Constructive
(24, 44, 517)-net in base 64, using
- base change [i] based on digital (13, 33, 517)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (2, 22, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- digital (1, 11, 258)-net over F256, using
- (u, u+v)-construction [i] based on
(44−20, 44, 2467)-Net over F64 — Digital
Digital (24, 44, 2467)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6444, 2467, F64, 20) (dual of [2467, 2423, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6444, 4113, F64, 20) (dual of [4113, 4069, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- 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(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(6444, 4113, F64, 20) (dual of [4113, 4069, 21]-code), using
(44−20, 44, 6365423)-Net in Base 64 — Upper bound on s
There is no (24, 44, 6365424)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 29 642785 585386 348190 465450 573643 353988 212868 411957 674515 409347 988072 161144 393751 > 6444 [i]