Best Known (59−10, 59, s)-Nets in Base 81
(59−10, 59, 3371758)-Net over F81 — Constructive and digital
Digital (49, 59, 3371758)-net over F81, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 5, 16318)-net over F81, using
- net defined by OOA [i] based on linear OOA(815, 16318, F81, 3, 3) (dual of [(16318, 3), 48949, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(815, 16318, F81, 2, 3) (dual of [(16318, 2), 32631, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(815, 16318, F81, 3, 3) (dual of [(16318, 3), 48949, 4]-NRT-code), using
- digital (12, 17, 1677720)-net over F81, using
- s-reduction based on digital (12, 17, 4194301)-net over F81, using
- net defined by OOA [i] based on linear OOA(8117, 4194301, F81, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(8117, 4194301, F81, 4, 5) (dual of [(4194301, 4), 16777187, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(8117, large, F81, 5) (dual of [large, large−17, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(8117, large, F81, 5) (dual of [large, large−17, 6]-code), using
- appending kth column [i] based on linear OOA(8117, 4194301, F81, 4, 5) (dual of [(4194301, 4), 16777187, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8117, 4194301, F81, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- s-reduction based on digital (12, 17, 4194301)-net over F81, using
- digital (27, 37, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8137, 1677720, F81, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(8137, 8388600, F81, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(8137, 8388600, F81, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(8137, 1677720, F81, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- digital (2, 5, 16318)-net over F81, using
(59−10, 59, large)-Net over F81 — Digital
Digital (49, 59, large)-net over F81, using
- t-expansion [i] based on digital (48, 59, large)-net over F81, using
- 6 times m-reduction [i] based on digital (48, 65, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- 6 times m-reduction [i] based on digital (48, 65, large)-net over F81, using
(59−10, 59, large)-Net in Base 81 — Upper bound on s
There is no (49, 59, large)-net in base 81, because
- 8 times m-reduction [i] would yield (49, 51, large)-net in base 81, but