Best Known (69−20, 69, s)-Nets in Base 9
(69−20, 69, 656)-Net over F9 — Constructive and digital
Digital (49, 69, 656)-net over F9, using
- net defined by OOA [i] based on linear OOA(969, 656, F9, 20, 20) (dual of [(656, 20), 13051, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(969, 6560, F9, 20) (dual of [6560, 6491, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(969, 6561, F9, 20) (dual of [6561, 6492, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(969, 6561, F9, 20) (dual of [6561, 6492, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(969, 6560, F9, 20) (dual of [6560, 6491, 21]-code), using
(69−20, 69, 3791)-Net over F9 — Digital
Digital (49, 69, 3791)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(969, 3791, F9, 20) (dual of [3791, 3722, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(969, 6561, F9, 20) (dual of [6561, 6492, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(969, 6561, F9, 20) (dual of [6561, 6492, 21]-code), using
(69−20, 69, 2173494)-Net in Base 9 — Upper bound on s
There is no (49, 69, 2173495)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 696201 221458 467974 912088 680432 206543 546457 928986 251118 746361 110065 > 969 [i]