Best Known (9, 9+4, s)-Nets in Base 81
(9, 9+4, 4194301)-Net over F81 — Constructive and digital
Digital (9, 13, 4194301)-net over F81, using
- net defined by OOA [i] based on linear OOA(8113, 4194301, F81, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(8113, 8388602, F81, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(8113, 8388602, F81, 4) (dual of [8388602, 8388589, 5]-code), using
(9, 9+4, large)-Net over F81 — Digital
Digital (9, 13, large)-net over F81, using
- net defined by OOA [i] based on linear OOA(8113, large, F81, 4, 4), using
- appending kth column [i] based on linear OOA(8113, large, F81, 3, 4), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- appending kth column [i] based on linear OOA(8113, large, F81, 3, 4), using
(9, 9+4, large)-Net in Base 81 — Upper bound on s
There is no (9, 13, large)-net in base 81, because
- 2 times m-reduction [i] would yield (9, 11, large)-net in base 81, but