Best Known (67, 67+15, s)-Nets in Base 81
(67, 67+15, 2396742)-Net over F81 — Constructive and digital
Digital (67, 82, 2396742)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (18, 25, 2796200)-net over F81, using
- net defined by OOA [i] based on linear OOA(8125, 2796200, F81, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8125, 8388601, F81, 7) (dual of [8388601, 8388576, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8125, 8388601, F81, 7) (dual of [8388601, 8388576, 8]-code), using
- net defined by OOA [i] based on linear OOA(8125, 2796200, F81, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
- digital (42, 57, 1198371)-net over F81, using
- net defined by OOA [i] based on linear OOA(8157, 1198371, F81, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8157, 8388598, F81, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8157, 8388598, F81, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(8157, 1198371, F81, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- digital (18, 25, 2796200)-net over F81, using
(67, 67+15, large)-Net over F81 — Digital
Digital (67, 82, large)-net over F81, using
- 811 times duplication [i] based on digital (66, 81, large)-net over F81, using
- t-expansion [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
- t-expansion [i] based on digital (60, 81, large)-net over F81, using
(67, 67+15, large)-Net in Base 81 — Upper bound on s
There is no (67, 82, large)-net in base 81, because
- 13 times m-reduction [i] would yield (67, 69, large)-net in base 81, but