Best Known (41, 56, s)-Nets in Base 81
(41, 56, 78107)-Net over F81 — Constructive and digital
Digital (41, 56, 78107)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 2187)-net over F81, using
- net defined by OOA [i] based on linear OOA(8113, 2187, F81, 7, 7) (dual of [(2187, 7), 15296, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8113, 6562, F81, 7) (dual of [6562, 6549, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, 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(8113, 6562, F81, 7) (dual of [6562, 6549, 8]-code), using
- net defined by OOA [i] based on linear OOA(8113, 2187, F81, 7, 7) (dual of [(2187, 7), 15296, 8]-NRT-code), using
- digital (28, 43, 75920)-net over F81, using
- net defined by OOA [i] based on linear OOA(8143, 75920, F81, 15, 15) (dual of [(75920, 15), 1138757, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using
- net defined by OOA [i] based on linear OOA(8143, 75920, F81, 15, 15) (dual of [(75920, 15), 1138757, 16]-NRT-code), using
- digital (6, 13, 2187)-net over F81, using
(41, 56, 3253185)-Net over F81 — Digital
Digital (41, 56, 3253185)-net over F81, using
(41, 56, large)-Net in Base 81 — Upper bound on s
There is no (41, 56, large)-net in base 81, because
- 13 times m-reduction [i] would yield (41, 43, large)-net in base 81, but