Best Known (40−20, 40, s)-Nets in Base 64
(40−20, 40, 410)-Net over F64 — Constructive and digital
Digital (20, 40, 410)-net over F64, using
- net defined by OOA [i] based on linear OOA(6440, 410, F64, 20, 20) (dual of [(410, 20), 8160, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6440, 4100, F64, 20) (dual of [4100, 4060, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6440, 4101, F64, 20) (dual of [4101, 4061, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [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(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(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(6440, 4101, F64, 20) (dual of [4101, 4061, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(6440, 4100, F64, 20) (dual of [4100, 4060, 21]-code), using
(40−20, 40, 514)-Net in Base 64 — Constructive
(20, 40, 514)-net in base 64, using
- base change [i] based on digital (10, 30, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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
- digital (0, 20, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 10, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(40−20, 40, 1367)-Net over F64 — Digital
Digital (20, 40, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6440, 1367, F64, 3, 20) (dual of [(1367, 3), 4061, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6440, 4101, F64, 20) (dual of [4101, 4061, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [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(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(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- OOA 3-folding [i] based on linear OA(6440, 4101, F64, 20) (dual of [4101, 4061, 21]-code), using
(40−20, 40, 1206018)-Net in Base 64 — Upper bound on s
There is no (20, 40, 1206019)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 1 766854 295432 221667 990359 801732 398114 373473 880785 353308 228566 208073 767840 > 6440 [i]