Best Known (88, 88+7, s)-Nets in Base 3
(88, 88+7, 8388600)-Net over F3 — Constructive and digital
Digital (88, 95, 8388600)-net over F3, using
- 32 times duplication [i] based on digital (86, 93, 8388600)-net over F3, using
- trace code for nets [i] based on 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
- trace code for nets [i] based on digital (24, 31, 2796200)-net over F27, using
(88, 88+7, large)-Net over F3 — Digital
Digital (88, 95, large)-net over F3, using
- 39 times duplication [i] based on digital (79, 86, large)-net over F3, using
- t-expansion [i] based on digital (78, 86, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(386, large, F3, 8) (dual of [large, large−86, 9]-code), using
- 10 times code embedding in larger space [i] based on linear OA(376, large, F3, 8) (dual of [large, large−76, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 10 times code embedding in larger space [i] based on linear OA(376, large, F3, 8) (dual of [large, large−76, 9]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(386, large, F3, 8) (dual of [large, large−86, 9]-code), using
- t-expansion [i] based on digital (78, 86, large)-net over F3, using
(88, 88+7, large)-Net in Base 3 — Upper bound on s
There is no (88, 95, large)-net in base 3, because
- 5 times m-reduction [i] would yield (88, 90, large)-net in base 3, but