Best Known (60, 60+12, s)-Nets in Base 81
(60, 60+12, 2799440)-Net over F81 — Constructive and digital
Digital (60, 72, 2799440)-net over F81, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 6, 3240)-net over F81, using
- net defined by OOA [i] based on linear OOA(816, 3240, F81, 4, 4) (dual of [(3240, 4), 12954, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(816, 6480, F81, 4) (dual of [6480, 6474, 5]-code), using
- 1 times truncation [i] based on linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(816, 6480, F81, 4) (dual of [6480, 6474, 5]-code), using
- net defined by OOA [i] based on linear OOA(816, 3240, F81, 4, 4) (dual of [(3240, 4), 12954, 5]-NRT-code), using
- digital (15, 21, 1398100)-net over F81, using
- s-reduction based on digital (15, 21, 2796201)-net over F81, using
- net defined by OOA [i] based on linear OOA(8121, 2796201, F81, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- net defined by OOA [i] based on linear OOA(8121, 2796201, F81, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
- s-reduction based on digital (15, 21, 2796201)-net over F81, using
- digital (33, 45, 1398100)-net over F81, using
- net defined by OOA [i] based on linear OOA(8145, 1398100, F81, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8145, 8388600, F81, 12) (dual of [8388600, 8388555, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(8145, 8388600, F81, 12) (dual of [8388600, 8388555, 13]-code), using
- net defined by OOA [i] based on linear OOA(8145, 1398100, F81, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- digital (2, 6, 3240)-net over F81, using
(60, 60+12, large)-Net over F81 — Digital
Digital (60, 72, large)-net over F81, using
- 9 times m-reduction [i] based on digital (60, 81, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
(60, 60+12, large)-Net in Base 81 — Upper bound on s
There is no (60, 72, large)-net in base 81, because
- 10 times m-reduction [i] would yield (60, 62, large)-net in base 81, but