Best Known (39−20, 39, s)-Nets in Base 64
(39−20, 39, 409)-Net over F64 — Constructive and digital
Digital (19, 39, 409)-net over F64, using
- net defined by OOA [i] based on linear OOA(6439, 409, F64, 20, 20) (dual of [(409, 20), 8141, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6439, 4090, F64, 20) (dual of [4090, 4051, 21]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(6439, 4090, F64, 20) (dual of [4090, 4051, 21]-code), using
(39−20, 39, 1243)-Net over F64 — Digital
Digital (19, 39, 1243)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6439, 1243, F64, 3, 20) (dual of [(1243, 3), 3690, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6439, 1366, F64, 3, 20) (dual of [(1366, 3), 4059, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6439, 4098, F64, 20) (dual of [4098, 4059, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [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(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- OOA 3-folding [i] based on linear OA(6439, 4098, F64, 20) (dual of [4098, 4059, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(6439, 1366, F64, 3, 20) (dual of [(1366, 3), 4059, 21]-NRT-code), using
(39−20, 39, 795673)-Net in Base 64 — Upper bound on s
There is no (19, 39, 795674)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 27607 021892 100896 690908 800976 217992 948625 658844 913591 858952 707001 194776 > 6439 [i]