Best Known (37−7, 37, s)-Nets in Base 27
(37−7, 37, 2816696)-Net over F27 — Constructive and digital
Digital (30, 37, 2816696)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (3, 6, 20496)-net over F27, using
- net defined by OOA [i] based on linear OOA(276, 20496, F27, 3, 3) (dual of [(20496, 3), 61482, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(276, 20496, F27, 2, 3) (dual of [(20496, 2), 40986, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(276, 20496, F27, 3, 3) (dual of [(20496, 3), 61482, 4]-NRT-code), using
- digital (24, 31, 2796200)-net over F27, using
- net defined by OOA [i] based on linear OOA(2731, 2796200, F27, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2731, 8388601, F27, 7) (dual of [8388601, 8388570, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2731, large, F27, 7) (dual of [large, large−31, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, 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(2731, large, F27, 7) (dual of [large, large−31, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2731, 8388601, F27, 7) (dual of [8388601, 8388570, 8]-code), using
- net defined by OOA [i] based on linear OOA(2731, 2796200, F27, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- digital (3, 6, 20496)-net over F27, using
(37−7, 37, large)-Net over F27 — Digital
Digital (30, 37, large)-net over F27, using
- 271 times duplication [i] based on digital (29, 36, large)-net over F27, using
- t-expansion [i] based on digital (28, 36, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2736, large, F27, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2736, large, F27, 8) (dual of [large, large−36, 9]-code), using
- t-expansion [i] based on digital (28, 36, large)-net over F27, using
(37−7, 37, large)-Net in Base 27 — Upper bound on s
There is no (30, 37, large)-net in base 27, because
- 5 times m-reduction [i] would yield (30, 32, large)-net in base 27, but