Best Known (74, 74+15, s)-Nets in Base 9
(74, 74+15, 151842)-Net over F9 — Constructive and digital
Digital (74, 89, 151842)-net over F9, using
- 91 times duplication [i] based on digital (73, 88, 151842)-net over F9, using
- net defined by OOA [i] based on linear OOA(988, 151842, F9, 15, 15) (dual of [(151842, 15), 2277542, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(988, 1062895, F9, 15) (dual of [1062895, 1062807, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(988, 1062898, F9, 15) (dual of [1062898, 1062810, 16]-code), using
- trace code [i] based on linear OA(8144, 531449, F81, 15) (dual of [531449, 531405, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(8143, 531442, F81, 15) (dual of [531442, 531399, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(8137, 531442, F81, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 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 C([0,7]) ⊂ C([0,6]) [i] based on
- trace code [i] based on linear OA(8144, 531449, F81, 15) (dual of [531449, 531405, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(988, 1062898, F9, 15) (dual of [1062898, 1062810, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(988, 1062895, F9, 15) (dual of [1062895, 1062807, 16]-code), using
- net defined by OOA [i] based on linear OOA(988, 151842, F9, 15, 15) (dual of [(151842, 15), 2277542, 16]-NRT-code), using
(74, 74+15, 1062900)-Net over F9 — Digital
Digital (74, 89, 1062900)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(989, 1062900, F9, 15) (dual of [1062900, 1062811, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(988, 1062898, F9, 15) (dual of [1062898, 1062810, 16]-code), using
- trace code [i] based on linear OA(8144, 531449, F81, 15) (dual of [531449, 531405, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(8143, 531442, F81, 15) (dual of [531442, 531399, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(8137, 531442, F81, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 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 C([0,7]) ⊂ C([0,6]) [i] based on
- trace code [i] based on linear OA(8144, 531449, F81, 15) (dual of [531449, 531405, 16]-code), using
- linear OA(988, 1062899, F9, 14) (dual of [1062899, 1062811, 15]-code), using Gilbert–Varšamov bound and bm = 988 > Vbs−1(k−1) = 195 096274 449505 849999 778866 197233 512863 310189 042254 841461 945377 633250 865364 810193 [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(988, 1062898, F9, 15) (dual of [1062898, 1062810, 16]-code), using
- construction X with Varšamov bound [i] based on
(74, 74+15, large)-Net in Base 9 — Upper bound on s
There is no (74, 89, large)-net in base 9, because
- 13 times m-reduction [i] would yield (74, 76, large)-net in base 9, but