Best Known (30, 30+7, s)-Nets in Base 3
(30, 30+7, 6561)-Net over F3 — Constructive and digital
Digital (30, 37, 6561)-net over F3, using
- net defined by OOA [i] based on linear OOA(337, 6561, F3, 7, 7) (dual of [(6561, 7), 45890, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(337, 19684, F3, 7) (dual of [19684, 19647, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, 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(337, 19684, F3, 7) (dual of [19684, 19647, 8]-code), using
(30, 30+7, 9842)-Net over F3 — Digital
Digital (30, 37, 9842)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(337, 9842, F3, 2, 7) (dual of [(9842, 2), 19647, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(337, 19684, F3, 7) (dual of [19684, 19647, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 2-folding [i] based on linear OA(337, 19684, F3, 7) (dual of [19684, 19647, 8]-code), using
(30, 30+7, 482843)-Net in Base 3 — Upper bound on s
There is no (30, 37, 482844)-net in base 3, because
- 1 times m-reduction [i] would yield (30, 36, 482844)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 150094 922066 455185 > 336 [i]