Best Known (28, 28+9, s)-Nets in Base 9
(28, 28+9, 3283)-Net over F9 — Constructive and digital
Digital (28, 37, 3283)-net over F9, using
- 91 times duplication [i] based on digital (27, 36, 3283)-net over F9, using
- net defined by OOA [i] based on linear OOA(936, 3283, F9, 9, 9) (dual of [(3283, 9), 29511, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(936, 13133, F9, 9) (dual of [13133, 13097, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(936, 13134, F9, 9) (dual of [13134, 13098, 10]-code), using
- trace code [i] based on linear OA(8118, 6567, F81, 9) (dual of [6567, 6549, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(8117, 6562, F81, 9) (dual of [6562, 6545, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(8113, 6562, F81, 7) (dual of [6562, 6549, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [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 C([0,4]) ⊂ C([0,3]) [i] based on
- trace code [i] based on linear OA(8118, 6567, F81, 9) (dual of [6567, 6549, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(936, 13134, F9, 9) (dual of [13134, 13098, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(936, 13133, F9, 9) (dual of [13133, 13097, 10]-code), using
- net defined by OOA [i] based on linear OOA(936, 3283, F9, 9, 9) (dual of [(3283, 9), 29511, 10]-NRT-code), using
(28, 28+9, 13136)-Net over F9 — Digital
Digital (28, 37, 13136)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(937, 13136, F9, 9) (dual of [13136, 13099, 10]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(936, 13134, F9, 9) (dual of [13134, 13098, 10]-code), using
- trace code [i] based on linear OA(8118, 6567, F81, 9) (dual of [6567, 6549, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(8117, 6562, F81, 9) (dual of [6562, 6545, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(8113, 6562, F81, 7) (dual of [6562, 6549, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [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 C([0,4]) ⊂ C([0,3]) [i] based on
- trace code [i] based on linear OA(8118, 6567, F81, 9) (dual of [6567, 6549, 10]-code), using
- linear OA(936, 13135, F9, 8) (dual of [13135, 13099, 9]-code), using Gilbert–Varšamov bound and bm = 936 > Vbs−1(k−1) = 28 009986 119251 897930 983990 464817 [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(936, 13134, F9, 9) (dual of [13134, 13098, 10]-code), using
- construction X with Varšamov bound [i] based on
(28, 28+9, large)-Net in Base 9 — Upper bound on s
There is no (28, 37, large)-net in base 9, because
- 7 times m-reduction [i] would yield (28, 30, large)-net in base 9, but