Best Known (57, 75, s)-Nets in Base 9
(57, 75, 1459)-Net over F9 — Constructive and digital
Digital (57, 75, 1459)-net over F9, using
- 1 times m-reduction [i] based on digital (57, 76, 1459)-net over F9, using
- net defined by OOA [i] based on linear OOA(976, 1459, F9, 19, 19) (dual of [(1459, 19), 27645, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(976, 13132, F9, 19) (dual of [13132, 13056, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(976, 13134, F9, 19) (dual of [13134, 13058, 20]-code), using
- trace code [i] based on linear OA(8138, 6567, F81, 19) (dual of [6567, 6529, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(8137, 6562, F81, 19) (dual of [6562, 6525, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- trace code [i] based on linear OA(8138, 6567, F81, 19) (dual of [6567, 6529, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(976, 13134, F9, 19) (dual of [13134, 13058, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(976, 13132, F9, 19) (dual of [13132, 13056, 20]-code), using
- net defined by OOA [i] based on linear OOA(976, 1459, F9, 19, 19) (dual of [(1459, 19), 27645, 20]-NRT-code), using
(57, 75, 14555)-Net over F9 — Digital
Digital (57, 75, 14555)-net over F9, using
(57, 75, large)-Net in Base 9 — Upper bound on s
There is no (57, 75, large)-net in base 9, because
- 16 times m-reduction [i] would yield (57, 59, large)-net in base 9, but