Best Known (11, 11+12, s)-Nets in Base 81
(11, 11+12, 1093)-Net over F81 — Constructive and digital
Digital (11, 23, 1093)-net over F81, using
- net defined by OOA [i] based on linear OOA(8123, 1093, F81, 12, 12) (dual of [(1093, 12), 13093, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8123, 6558, F81, 12) (dual of [6558, 6535, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(8123, 6558, F81, 12) (dual of [6558, 6535, 13]-code), using
(11, 11+12, 2187)-Net over F81 — Digital
Digital (11, 23, 2187)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8123, 2187, F81, 3, 12) (dual of [(2187, 3), 6538, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- OOA 3-folding [i] based on linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using
(11, 11+12, 774443)-Net in Base 81 — Upper bound on s
There is no (11, 23, 774444)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 78 551811 729639 683562 889236 239347 104517 973121 > 8123 [i]