Best Known (52, 73, s)-Nets in Base 9
(52, 73, 656)-Net over F9 — Constructive and digital
Digital (52, 73, 656)-net over F9, using
- net defined by OOA [i] based on linear OOA(973, 656, F9, 21, 21) (dual of [(656, 21), 13703, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(973, 6561, F9, 21) (dual of [6561, 6488, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 10-folding and stacking with additional row [i] based on linear OA(973, 6561, F9, 21) (dual of [6561, 6488, 22]-code), using
(52, 73, 4084)-Net over F9 — Digital
Digital (52, 73, 4084)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(973, 4084, F9, 21) (dual of [4084, 4011, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(973, 6561, F9, 21) (dual of [6561, 6488, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(973, 6561, F9, 21) (dual of [6561, 6488, 22]-code), using
(52, 73, 4201765)-Net in Base 9 — Upper bound on s
There is no (52, 73, 4201766)-net in base 9, because
- 1 times m-reduction [i] would yield (52, 72, 4201766)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 507 529634 091628 238241 350036 326375 467432 743148 356706 256458 867690 540065 > 972 [i]