Best Known (31, 37, s)-Nets in Base 27
(31, 37, 5592430)-Net over F27 — Constructive and digital
Digital (31, 37, 5592430)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 28)-net over F27, using
- digital (6, 9, 2796201)-net over F27, using
- s-reduction based on digital (6, 9, large)-net over F27, using
- net defined by OOA [i] based on linear OOA(279, large, F27, 3, 3), using
- appending kth column [i] based on linear OOA(279, large, F27, 2, 3), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(279, large, F27, 3) (dual of [large, large−9, 4]-code), using
- appending kth column [i] based on linear OOA(279, large, F27, 2, 3), using
- net defined by OOA [i] based on linear OOA(279, large, F27, 3, 3), using
- s-reduction based on digital (6, 9, large)-net over F27, using
- digital (20, 26, 2796201)-net over F27, using
- net defined by OOA [i] based on linear OOA(2726, 2796201, F27, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2726, large, F27, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(2726, large, F27, 6) (dual of [large, large−26, 7]-code), using
- net defined by OOA [i] based on linear OOA(2726, 2796201, F27, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
(31, 37, large)-Net over F27 — Digital
Digital (31, 37, large)-net over F27, using
- 271 times duplication [i] based on digital (30, 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
(31, 37, large)-Net in Base 27 — Upper bound on s
There is no (31, 37, large)-net in base 27, because
- 4 times m-reduction [i] would yield (31, 33, large)-net in base 27, but