Best Known (44, 44+9, s)-Nets in Base 9
(44, 44+9, 265724)-Net over F9 — Constructive and digital
Digital (44, 53, 265724)-net over F9, using
- 91 times duplication [i] based on digital (43, 52, 265724)-net over F9, using
- net defined by OOA [i] based on linear OOA(952, 265724, F9, 9, 9) (dual of [(265724, 9), 2391464, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(952, 1062897, F9, 9) (dual of [1062897, 1062845, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(952, 1062898, F9, 9) (dual of [1062898, 1062846, 10]-code), using
- trace code [i] based on linear OA(8126, 531449, F81, 9) (dual of [531449, 531423, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(8125, 531442, F81, 9) (dual of [531442, 531417, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(8119, 531442, F81, 7) (dual of [531442, 531423, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- trace code [i] based on linear OA(8126, 531449, F81, 9) (dual of [531449, 531423, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(952, 1062898, F9, 9) (dual of [1062898, 1062846, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(952, 1062897, F9, 9) (dual of [1062897, 1062845, 10]-code), using
- net defined by OOA [i] based on linear OOA(952, 265724, F9, 9, 9) (dual of [(265724, 9), 2391464, 10]-NRT-code), using
(44, 44+9, 1062900)-Net over F9 — Digital
Digital (44, 53, 1062900)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(953, 1062900, F9, 9) (dual of [1062900, 1062847, 10]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(952, 1062898, F9, 9) (dual of [1062898, 1062846, 10]-code), using
- trace code [i] based on linear OA(8126, 531449, F81, 9) (dual of [531449, 531423, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(8125, 531442, F81, 9) (dual of [531442, 531417, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(8119, 531442, F81, 7) (dual of [531442, 531423, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- trace code [i] based on linear OA(8126, 531449, F81, 9) (dual of [531449, 531423, 10]-code), using
- linear OA(952, 1062899, F9, 8) (dual of [1062899, 1062847, 9]-code), using Gilbert–Varšamov bound and bm = 952 > Vbs−1(k−1) = 637 723290 906687 576572 761467 733912 788063 173073 [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(952, 1062898, F9, 9) (dual of [1062898, 1062846, 10]-code), using
- construction X with Varšamov bound [i] based on
(44, 44+9, large)-Net in Base 9 — Upper bound on s
There is no (44, 53, large)-net in base 9, because
- 7 times m-reduction [i] would yield (44, 46, large)-net in base 9, but