Best Known (53−7, 53, s)-Nets in Base 3
(53−7, 53, 531441)-Net over F3 — Constructive and digital
Digital (46, 53, 531441)-net over F3, using
- net defined by OOA [i] based on linear OOA(353, 531441, F3, 7, 7) (dual of [(531441, 7), 3720034, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(353, 1594324, F3, 7) (dual of [1594324, 1594271, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, 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(353, 1594324, F3, 7) (dual of [1594324, 1594271, 8]-code), using
(53−7, 53, 797162)-Net over F3 — Digital
Digital (46, 53, 797162)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(353, 797162, F3, 2, 7) (dual of [(797162, 2), 1594271, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(353, 1594324, F3, 7) (dual of [1594324, 1594271, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, 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(353, 1594324, F3, 7) (dual of [1594324, 1594271, 8]-code), using
(53−7, 53, large)-Net in Base 3 — Upper bound on s
There is no (46, 53, large)-net in base 3, because
- 5 times m-reduction [i] would yield (46, 48, large)-net in base 3, but