Best Known (41, 41+10, s)-Nets in Base 81
(41, 41+10, 1943444)-Net over F81 — Constructive and digital
Digital (41, 51, 1943444)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (9, 14, 265724)-net over F81, using
- net defined by OOA [i] based on linear OOA(8114, 265724, F81, 5, 5) (dual of [(265724, 5), 1328606, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(8114, 265724, F81, 4, 5) (dual of [(265724, 4), 1062882, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(8114, 531449, F81, 5) (dual of [531449, 531435, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(8113, 531442, F81, 5) (dual of [531442, 531429, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(817, 531442, F81, 3) (dual of [531442, 531435, 4]-code or 531442-cap in PG(6,81)), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(8114, 531449, F81, 5) (dual of [531449, 531435, 6]-code), using
- appending kth column [i] based on linear OOA(8114, 265724, F81, 4, 5) (dual of [(265724, 4), 1062882, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8114, 265724, F81, 5, 5) (dual of [(265724, 5), 1328606, 6]-NRT-code), 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 (9, 14, 265724)-net over F81, using
(41, 41+10, large)-Net over F81 — Digital
Digital (41, 51, large)-net over F81, using
- t-expansion [i] based on digital (39, 51, large)-net over F81, using
- 2 times m-reduction [i] based on digital (39, 53, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- 2 times m-reduction [i] based on digital (39, 53, large)-net over F81, using
(41, 41+10, large)-Net in Base 81 — Upper bound on s
There is no (41, 51, large)-net in base 81, because
- 8 times m-reduction [i] would yield (41, 43, large)-net in base 81, but