Best Known (31, 31+8, s)-Nets in Base 27
(31, 31+8, 2097150)-Net over F27 — Constructive and digital
Digital (31, 39, 2097150)-net over F27, using
- 273 times duplication [i] based on digital (28, 36, 2097150)-net over F27, using
- net defined by OOA [i] based on linear OOA(2736, 2097150, F27, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2736, 8388600, F27, 8) (dual of [8388600, 8388564, 9]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2736, large, F27, 8) (dual of [large, large−36, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2736, 8388600, F27, 8) (dual of [8388600, 8388564, 9]-code), using
- net defined by OOA [i] based on linear OOA(2736, 2097150, F27, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
(31, 31+8, large)-Net over F27 — Digital
Digital (31, 39, large)-net over F27, using
- 273 times duplication [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
(31, 31+8, large)-Net in Base 27 — Upper bound on s
There is no (31, 39, large)-net in base 27, because
- 6 times m-reduction [i] would yield (31, 33, large)-net in base 27, but