Best Known (20, 20+21, s)-Nets in Base 64
(20, 20+21, 409)-Net over F64 — Constructive and digital
Digital (20, 41, 409)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 409, F64, 21, 21) (dual of [(409, 21), 8548, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6441, 4091, F64, 21) (dual of [4091, 4050, 22]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(6441, 4096, F64, 21) (dual of [4096, 4055, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6441, 4091, F64, 21) (dual of [4091, 4050, 22]-code), using
(20, 20+21, 1234)-Net over F64 — Digital
Digital (20, 41, 1234)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6441, 1234, F64, 3, 21) (dual of [(1234, 3), 3661, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6441, 1366, F64, 3, 21) (dual of [(1366, 3), 4057, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6441, 4098, F64, 21) (dual of [4098, 4057, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- 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(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(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- OOA 3-folding [i] based on linear OA(6441, 4098, F64, 21) (dual of [4098, 4057, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(6441, 1366, F64, 3, 21) (dual of [(1366, 3), 4057, 22]-NRT-code), using
(20, 20+21, 1206018)-Net in Base 64 — Upper bound on s
There is no (20, 41, 1206019)-net in base 64, because
- 1 times m-reduction [i] would yield (20, 40, 1206019)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 1 766854 295432 221667 990359 801732 398114 373473 880785 353308 228566 208073 767840 > 6440 [i]