Best Known (9, 9+10, s)-Nets in Base 64
(9, 9+10, 819)-Net over F64 — Constructive and digital
Digital (9, 19, 819)-net over F64, using
- net defined by OOA [i] based on linear OOA(6419, 819, F64, 10, 10) (dual of [(819, 10), 8171, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6419, 4095, F64, 10) (dual of [4095, 4076, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6419, 4095, F64, 10) (dual of [4095, 4076, 11]-code), using
(9, 9+10, 1366)-Net over F64 — Digital
Digital (9, 19, 1366)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6419, 1366, F64, 3, 10) (dual of [(1366, 3), 4079, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6419, 4098, F64, 10) (dual of [4098, 4079, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- OOA 3-folding [i] based on linear OA(6419, 4098, F64, 10) (dual of [4098, 4079, 11]-code), using
(9, 9+10, 301978)-Net in Base 64 — Upper bound on s
There is no (9, 19, 301979)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 20769 219038 736778 615683 634524 265804 > 6419 [i]