Best Known (41, 41+15, s)-Nets in Base 27
(41, 41+15, 3055)-Net over F27 — Constructive and digital
Digital (41, 56, 3055)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 243)-net over F27, using
- net defined by OOA [i] based on linear OOA(2713, 243, F27, 7, 7) (dual of [(243, 7), 1688, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2713, 730, F27, 7) (dual of [730, 717, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(2713, 730, F27, 7) (dual of [730, 717, 8]-code), using
- net defined by OOA [i] based on linear OOA(2713, 243, F27, 7, 7) (dual of [(243, 7), 1688, 8]-NRT-code), using
- digital (28, 43, 2812)-net over F27, using
- net defined by OOA [i] based on linear OOA(2743, 2812, F27, 15, 15) (dual of [(2812, 15), 42137, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2743, 19685, F27, 15) (dual of [19685, 19642, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2743, 19686, F27, 15) (dual of [19686, 19643, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(2743, 19683, F27, 15) (dual of [19683, 19640, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2740, 19683, F27, 14) (dual of [19683, 19643, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(270, 3, F27, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(2743, 19686, F27, 15) (dual of [19686, 19643, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2743, 19685, F27, 15) (dual of [19685, 19642, 16]-code), using
- net defined by OOA [i] based on linear OOA(2743, 2812, F27, 15, 15) (dual of [(2812, 15), 42137, 16]-NRT-code), using
- digital (6, 13, 243)-net over F27, using
(41, 41+15, 123584)-Net over F27 — Digital
Digital (41, 56, 123584)-net over F27, using
(41, 41+15, large)-Net in Base 27 — Upper bound on s
There is no (41, 56, large)-net in base 27, because
- 13 times m-reduction [i] would yield (41, 43, large)-net in base 27, but