Best Known (18−5, 18, s)-Nets in Base 81
(18−5, 18, 4194301)-Net over F81 — Constructive and digital
Digital (13, 18, 4194301)-net over F81, using
- 811 times duplication [i] 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
(18−5, 18, large)-Net over F81 — Digital
Digital (13, 18, large)-net over F81, using
- 811 times duplication [i] based on digital (12, 17, large)-net over F81, using
- net defined by OOA [i] based on linear OOA(8117, large, F81, 5, 5), using
- appending kth column [i] based on linear OOA(8117, large, F81, 4, 5), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [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, large, F81, 4, 5), using
- net defined by OOA [i] based on linear OOA(8117, large, F81, 5, 5), using
(18−5, 18, large)-Net in Base 81 — Upper bound on s
There is no (13, 18, large)-net in base 81, because
- 3 times m-reduction [i] would yield (13, 15, large)-net in base 81, but