Best Known (44, 44+15, s)-Nets in Base 9
(44, 44+15, 1875)-Net over F9 — Constructive and digital
Digital (44, 59, 1875)-net over F9, using
- 91 times duplication [i] based on 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
(44, 44+15, 12808)-Net over F9 — Digital
Digital (44, 59, 12808)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(959, 12808, F9, 15) (dual of [12808, 12749, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(959, 13127, F9, 15) (dual of [13127, 13068, 16]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(958, 13126, F9, 15) (dual of [13126, 13068, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(959, 13127, F9, 15) (dual of [13127, 13068, 16]-code), using
(44, 44+15, large)-Net in Base 9 — Upper bound on s
There is no (44, 59, large)-net in base 9, because
- 13 times m-reduction [i] would yield (44, 46, large)-net in base 9, but