Best Known (54, 72, s)-Nets in Base 81
(54, 72, 932067)-Net over F81 — Constructive and digital
Digital (54, 72, 932067)-net over F81, using
- 813 times duplication [i] based on digital (51, 69, 932067)-net over F81, using
- net defined by OOA [i] based on linear OOA(8169, 932067, F81, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- OA 9-folding and stacking [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]
- OA 9-folding and stacking [i] based on linear OA(8169, large, F81, 18) (dual of [large, large−69, 19]-code), using
- net defined by OOA [i] based on linear OOA(8169, 932067, F81, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
(54, 72, large)-Net over F81 — Digital
Digital (54, 72, large)-net over F81, using
- 1 times m-reduction [i] based on digital (54, 73, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8173, large, F81, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8173, large, F81, 19) (dual of [large, large−73, 20]-code), using
(54, 72, large)-Net in Base 81 — Upper bound on s
There is no (54, 72, large)-net in base 81, because
- 16 times m-reduction [i] would yield (54, 56, large)-net in base 81, but