Best Known (43−20, 43, s)-Nets in Base 64
(43−20, 43, 411)-Net over F64 — Constructive and digital
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
(43−20, 43, 516)-Net in Base 64 — Constructive
(23, 43, 516)-net in base 64, using
- 1 times m-reduction [i] based on (23, 44, 516)-net in base 64, using
- base change [i] based on digital (12, 33, 516)-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 (1, 22, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 11, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (12, 33, 516)-net over F256, using
(43−20, 43, 2055)-Net over F64 — Digital
Digital (23, 43, 2055)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6443, 2055, F64, 2, 20) (dual of [(2055, 2), 4067, 21]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(6443, 4110, F64, 20) (dual of [4110, 4067, 21]-code), using
(43−20, 43, 4199611)-Net in Base 64 — Upper bound on s
There is no (23, 43, 4199612)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 463168 440293 089729 123820 025248 505147 452742 402485 992522 179046 604241 642947 795233 > 6443 [i]