Best Known (40, 40+15, s)-Nets in Base 27
(40, 40+15, 2896)-Net over F27 — Constructive and digital
Digital (40, 55, 2896)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (5, 12, 84)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 28)-net over F27, using
- digital (0, 3, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 7, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- generalized (u, u+v)-construction [i] based on
- 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 (5, 12, 84)-net over F27, using
(40, 40+15, 2928)-Net in Base 27 — Constructive
(40, 55, 2928)-net in base 27, using
- (u, u+v)-construction [i] based on
- (5, 12, 116)-net in base 27, using
- base change [i] based on digital (2, 9, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- base change [i] based on digital (2, 9, 116)-net over F81, 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
- (5, 12, 116)-net in base 27, using
(40, 40+15, 97663)-Net over F27 — Digital
Digital (40, 55, 97663)-net over F27, using
(40, 40+15, large)-Net in Base 27 — Upper bound on s
There is no (40, 55, large)-net in base 27, because
- 13 times m-reduction [i] would yield (40, 42, large)-net in base 27, but