Best Known (40, 40+14, s)-Nets in Base 9
(40, 40+14, 1875)-Net over F9 — Constructive and digital
Digital (40, 54, 1875)-net over F9, using
- net defined by OOA [i] based on linear OOA(954, 1875, F9, 14, 14) (dual of [(1875, 14), 26196, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(954, 13125, F9, 14) (dual of [13125, 13071, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(954, 13126, F9, 14) (dual of [13126, 13072, 15]-code), using
- trace code [i] based on linear OA(8127, 6563, F81, 14) (dual of [6563, 6536, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 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(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(8127, 6563, F81, 14) (dual of [6563, 6536, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(954, 13126, F9, 14) (dual of [13126, 13072, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(954, 13125, F9, 14) (dual of [13125, 13071, 15]-code), using
(40, 40+14, 10829)-Net over F9 — Digital
Digital (40, 54, 10829)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(954, 10829, F9, 14) (dual of [10829, 10775, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(954, 13122, F9, 14) (dual of [13122, 13068, 15]-code), using
- trace code [i] based on 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]
- trace code [i] based on linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(954, 13122, F9, 14) (dual of [13122, 13068, 15]-code), using
(40, 40+14, large)-Net in Base 9 — Upper bound on s
There is no (40, 54, large)-net in base 9, because
- 12 times m-reduction [i] would yield (40, 42, large)-net in base 9, but