Best Known (57−14, 57, s)-Nets in Base 9
(57−14, 57, 1876)-Net over F9 — Constructive and digital
Digital (43, 57, 1876)-net over F9, using
- 91 times duplication [i] based on digital (42, 56, 1876)-net over F9, using
- net defined by OOA [i] based on linear OOA(956, 1876, F9, 14, 14) (dual of [(1876, 14), 26208, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(956, 13132, F9, 14) (dual of [13132, 13076, 15]-code), using
- trace code [i] based on linear OA(8128, 6566, F81, 14) (dual of [6566, 6538, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- 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(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(8128, 6566, F81, 14) (dual of [6566, 6538, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(956, 13132, F9, 14) (dual of [13132, 13076, 15]-code), using
- net defined by OOA [i] based on linear OOA(956, 1876, F9, 14, 14) (dual of [(1876, 14), 26208, 15]-NRT-code), using
(57−14, 57, 13134)-Net over F9 — Digital
Digital (43, 57, 13134)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(957, 13134, F9, 14) (dual of [13134, 13077, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(956, 13132, F9, 14) (dual of [13132, 13076, 15]-code), using
- trace code [i] based on linear OA(8128, 6566, F81, 14) (dual of [6566, 6538, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- 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(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(8128, 6566, F81, 14) (dual of [6566, 6538, 15]-code), using
- linear OA(956, 13133, F9, 13) (dual of [13133, 13077, 14]-code), using Gilbert–Varšamov bound and bm = 956 > Vbs−1(k−1) = 3754 738238 121497 291601 696007 680471 667282 878101 727969 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(956, 13132, F9, 14) (dual of [13132, 13076, 15]-code), using
- construction X with Varšamov bound [i] based on
(57−14, 57, large)-Net in Base 9 — Upper bound on s
There is no (43, 57, large)-net in base 9, because
- 12 times m-reduction [i] would yield (43, 45, large)-net in base 9, but