Best Known (46, 65, s)-Nets in Base 9
(46, 65, 729)-Net over F9 — Constructive and digital
Digital (46, 65, 729)-net over F9, using
- net defined by OOA [i] based on linear OOA(965, 729, F9, 19, 19) (dual of [(729, 19), 13786, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(965, 6562, F9, 19) (dual of [6562, 6497, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(965, 6562, F9, 19) (dual of [6562, 6497, 20]-code), using
(46, 65, 3500)-Net over F9 — Digital
Digital (46, 65, 3500)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(965, 3500, F9, 19) (dual of [3500, 3435, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(965, 6561, F9, 19) (dual of [6561, 6496, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(965, 6561, F9, 19) (dual of [6561, 6496, 20]-code), using
(46, 65, 3165084)-Net in Base 9 — Upper bound on s
There is no (46, 65, 3165085)-net in base 9, because
- 1 times m-reduction [i] would yield (46, 64, 3165085)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 11 790211 029368 855271 939822 408735 003403 023937 231295 647676 802857 > 964 [i]