Best Known (58−14, 58, s)-Nets in Base 9
(58−14, 58, 1876)-Net over F9 — Constructive and digital
Digital (44, 58, 1876)-net over F9, using
- 92 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
(58−14, 58, 13138)-Net over F9 — Digital
Digital (44, 58, 13138)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(958, 13138, F9, 14) (dual of [13138, 13080, 15]-code), using
- trace code [i] based on linear OA(8129, 6569, F81, 14) (dual of [6569, 6540, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [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(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- trace code [i] based on linear OA(8129, 6569, F81, 14) (dual of [6569, 6540, 15]-code), using
(58−14, 58, large)-Net in Base 9 — Upper bound on s
There is no (44, 58, large)-net in base 9, because
- 12 times m-reduction [i] would yield (44, 46, large)-net in base 9, but