Best Known (42−7, 42, s)-Nets in Base 27
(42−7, 42, 5592428)-Net over F27 — Constructive and digital
Digital (35, 42, 5592428)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 28)-net over F27, using
- digital (6, 9, 2796200)-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 (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
(42−7, 42, large)-Net over F27 — Digital
Digital (35, 42, large)-net over F27, using
- 271 times duplication [i] based on digital (34, 41, large)-net over F27, using
- t-expansion [i] based on digital (32, 41, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- t-expansion [i] based on digital (32, 41, large)-net over F27, using
(42−7, 42, large)-Net in Base 27 — Upper bound on s
There is no (35, 42, large)-net in base 27, because
- 5 times m-reduction [i] would yield (35, 37, large)-net in base 27, but