Best Known (19−7, 19, s)-Nets in Base 27
(19−7, 19, 6561)-Net over F27 — Constructive and digital
Digital (12, 19, 6561)-net over F27, using
- net defined by OOA [i] based on linear OOA(2719, 6561, F27, 7, 7) (dual of [(6561, 7), 45908, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2719, 19684, F27, 7) (dual of [19684, 19665, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(2719, 19684, F27, 7) (dual of [19684, 19665, 8]-code), using
(19−7, 19, 14247)-Net over F27 — Digital
Digital (12, 19, 14247)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2719, 14247, F27, 7) (dual of [14247, 14228, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2719, 19683, F27, 7) (dual of [19683, 19664, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(2719, 19683, F27, 7) (dual of [19683, 19664, 8]-code), using
(19−7, 19, large)-Net in Base 27 — Upper bound on s
There is no (12, 19, large)-net in base 27, because
- 5 times m-reduction [i] would yield (12, 14, large)-net in base 27, but