Best Known (32−9, 32, s)-Nets in Base 9
(32−9, 32, 1639)-Net over F9 — Constructive and digital
Digital (23, 32, 1639)-net over F9, using
- net defined by OOA [i] based on linear OOA(932, 1639, F9, 9, 9) (dual of [(1639, 9), 14719, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(932, 6557, F9, 9) (dual of [6557, 6525, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(932, 6560, F9, 9) (dual of [6560, 6528, 10]-code), using
- 1 times truncation [i] based on linear OA(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times truncation [i] based on linear OA(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(932, 6560, F9, 9) (dual of [6560, 6528, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(932, 6557, F9, 9) (dual of [6557, 6525, 10]-code), using
(32−9, 32, 6560)-Net over F9 — Digital
Digital (23, 32, 6560)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(932, 6560, F9, 9) (dual of [6560, 6528, 10]-code), using
- 1 times truncation [i] based on linear OA(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times truncation [i] based on linear OA(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using
(32−9, 32, 6876099)-Net in Base 9 — Upper bound on s
There is no (23, 32, 6876100)-net in base 9, because
- 1 times m-reduction [i] would yield (23, 31, 6876100)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 381520 560783 387407 094010 390401 > 931 [i]