Best Known (26, 26+7, s)-Nets in Base 3
(26, 26+7, 2187)-Net over F3 — Constructive and digital
Digital (26, 33, 2187)-net over F3, using
- net defined by OOA [i] based on linear OOA(333, 2187, F3, 7, 7) (dual of [(2187, 7), 15276, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(333, 6562, F3, 7) (dual of [6562, 6529, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, 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(333, 6562, F3, 7) (dual of [6562, 6529, 8]-code), using
(26, 26+7, 3281)-Net over F3 — Digital
Digital (26, 33, 3281)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(333, 3281, F3, 2, 7) (dual of [(3281, 2), 6529, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(333, 6562, F3, 7) (dual of [6562, 6529, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, 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(333, 6562, F3, 7) (dual of [6562, 6529, 8]-code), using
(26, 26+7, 111593)-Net in Base 3 — Upper bound on s
There is no (26, 33, 111594)-net in base 3, because
- 1 times m-reduction [i] would yield (26, 32, 111594)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1853 064168 475185 > 332 [i]