Best Known (66−14, 66, s)-Nets in Base 81
(66−14, 66, 1200558)-Net over F81 — Constructive and digital
Digital (52, 66, 1200558)-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 (39, 53, 1198371)-net over F81, using
- net defined by OOA [i] based on linear OOA(8153, 1198371, F81, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(8153, 8388597, F81, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(8153, 8388597, F81, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(8153, 1198371, F81, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- digital (6, 13, 2187)-net over F81, using
(66−14, 66, large)-Net over F81 — Digital
Digital (52, 66, large)-net over F81, using
- t-expansion [i] based on digital (51, 66, large)-net over F81, using
- 3 times m-reduction [i] based on digital (51, 69, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8169, large, F81, 18) (dual of [large, large−69, 19]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8169, large, F81, 18) (dual of [large, large−69, 19]-code), using
- 3 times m-reduction [i] based on digital (51, 69, large)-net over F81, using
(66−14, 66, large)-Net in Base 81 — Upper bound on s
There is no (52, 66, large)-net in base 81, because
- 12 times m-reduction [i] would yield (52, 54, large)-net in base 81, but