Best Known (9, 9+10, s)-Nets in Base 81
(9, 9+10, 1312)-Net over F81 — Constructive and digital
Digital (9, 19, 1312)-net over F81, using
- net defined by OOA [i] based on linear OOA(8119, 1312, F81, 10, 10) (dual of [(1312, 10), 13101, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(8119, 6560, F81, 10) (dual of [6560, 6541, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(8119, 6560, F81, 10) (dual of [6560, 6541, 11]-code), using
(9, 9+10, 2187)-Net over F81 — Digital
Digital (9, 19, 2187)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8119, 2187, F81, 3, 10) (dual of [(2187, 3), 6542, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- OOA 3-folding [i] based on linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using
(9, 9+10, 582086)-Net in Base 81 — Upper bound on s
There is no (9, 19, 582087)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 1 824802 568552 487490 593136 634571 906801 > 8119 [i]