Best Known (43, 43+15, s)-Nets in Base 9
(43, 43+15, 1875)-Net over F9 — Constructive and digital
Digital (43, 58, 1875)-net over F9, using
- net defined by OOA [i] based on linear OOA(958, 1875, F9, 15, 15) (dual of [(1875, 15), 28067, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(958, 13126, F9, 15) (dual of [13126, 13068, 16]-code), using
- trace code [i] based on linear OA(8129, 6563, F81, 15) (dual of [6563, 6534, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(8129, 6561, F81, 15) (dual of [6561, 6532, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- trace code [i] based on linear OA(8129, 6563, F81, 15) (dual of [6563, 6534, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(958, 13126, F9, 15) (dual of [13126, 13068, 16]-code), using
(43, 43+15, 10815)-Net over F9 — Digital
Digital (43, 58, 10815)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(958, 10815, F9, 15) (dual of [10815, 10757, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(958, 13124, F9, 15) (dual of [13124, 13066, 16]-code), using
- trace code [i] based on linear OA(8129, 6562, F81, 15) (dual of [6562, 6533, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- trace code [i] based on linear OA(8129, 6562, F81, 15) (dual of [6562, 6533, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(958, 13124, F9, 15) (dual of [13124, 13066, 16]-code), using
(43, 43+15, large)-Net in Base 9 — Upper bound on s
There is no (43, 58, large)-net in base 9, because
- 13 times m-reduction [i] would yield (43, 45, large)-net in base 9, but