Best Known (50, 65, s)-Nets in Base 9
(50, 65, 1885)-Net over F9 — Constructive and digital
Digital (50, 65, 1885)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 10)-net over F9, using
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 0 and N(F) ≥ 10, using
- the rational function field F9(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- digital (43, 58, 1875)-net over F9, using
- net defined by OOA [i] based on linear OOA(958, 1875, F9, 15, 15) (dual of [(1875, 15), 28067, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(958, 13126, F9, 15) (dual of [13126, 13068, 16]-code), using
- trace code [i] based on linear OA(8129, 6563, F81, 15) (dual of [6563, 6534, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(8129, 6561, F81, 15) (dual of [6561, 6532, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- trace code [i] based on linear OA(8129, 6563, F81, 15) (dual of [6563, 6534, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(958, 13126, F9, 15) (dual of [13126, 13068, 16]-code), using
- net defined by OOA [i] based on linear OOA(958, 1875, F9, 15, 15) (dual of [(1875, 15), 28067, 16]-NRT-code), using
- digital (0, 7, 10)-net over F9, using
(50, 65, 2812)-Net in Base 9 — Constructive
(50, 65, 2812)-net in base 9, using
- net defined by OOA [i] based on OOA(965, 2812, S9, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(965, 19685, S9, 15), using
- discarding factors based on OA(965, 19686, S9, 15), using
- discarding parts of the base [i] 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 parts of the base [i] based on linear OA(2743, 19686, F27, 15) (dual of [19686, 19643, 16]-code), using
- discarding factors based on OA(965, 19686, S9, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(965, 19685, S9, 15), using
(50, 65, 20367)-Net over F9 — Digital
Digital (50, 65, 20367)-net over F9, using
(50, 65, large)-Net in Base 9 — Upper bound on s
There is no (50, 65, large)-net in base 9, because
- 13 times m-reduction [i] would yield (50, 52, large)-net in base 9, but