Best Known (18, 18+7, s)-Nets in Base 3
(18, 18+7, 243)-Net over F3 — Constructive and digital
Digital (18, 25, 243)-net over F3, using
- net defined by OOA [i] based on linear OOA(325, 243, F3, 7, 7) (dual of [(243, 7), 1676, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(325, 730, F3, 7) (dual of [730, 705, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, 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(325, 730, F3, 7) (dual of [730, 705, 8]-code), using
(18, 18+7, 365)-Net over F3 — Digital
Digital (18, 25, 365)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(325, 365, F3, 2, 7) (dual of [(365, 2), 705, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(325, 730, F3, 7) (dual of [730, 705, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, 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(325, 730, F3, 7) (dual of [730, 705, 8]-code), using
(18, 18+7, 5958)-Net in Base 3 — Upper bound on s
There is no (18, 25, 5959)-net in base 3, because
- 1 times m-reduction [i] would yield (18, 24, 5959)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 282491 436315 > 324 [i]