Best Known (33−8, 33, s)-Nets in Base 9
(33−8, 33, 3283)-Net over F9 — Constructive and digital
Digital (25, 33, 3283)-net over F9, using
- 91 times duplication [i] based on digital (24, 32, 3283)-net over F9, using
- net defined by OOA [i] based on linear OOA(932, 3283, F9, 8, 8) (dual of [(3283, 8), 26232, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(932, 13132, F9, 8) (dual of [13132, 13100, 9]-code), using
- trace code [i] based on linear OA(8116, 6566, F81, 8) (dual of [6566, 6550, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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 Ce(7) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(8116, 6566, F81, 8) (dual of [6566, 6550, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(932, 13132, F9, 8) (dual of [13132, 13100, 9]-code), using
- net defined by OOA [i] based on linear OOA(932, 3283, F9, 8, 8) (dual of [(3283, 8), 26232, 9]-NRT-code), using
(33−8, 33, 4921)-Net in Base 9 — Constructive
(25, 33, 4921)-net in base 9, using
- base change [i] based on digital (14, 22, 4921)-net over F27, using
- net defined by OOA [i] based on linear OOA(2722, 4921, F27, 8, 8) (dual of [(4921, 8), 39346, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2722, 19684, F27, 8) (dual of [19684, 19662, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2722, 19686, F27, 8) (dual of [19686, 19664, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(2722, 19683, F27, 8) (dual of [19683, 19661, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2719, 19683, F27, 7) (dual of [19683, 19664, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(270, 3, F27, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(2722, 19686, F27, 8) (dual of [19686, 19664, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2722, 19684, F27, 8) (dual of [19684, 19662, 9]-code), using
- net defined by OOA [i] based on linear OOA(2722, 4921, F27, 8, 8) (dual of [(4921, 8), 39346, 9]-NRT-code), using
(33−8, 33, 13320)-Net over F9 — Digital
Digital (25, 33, 13320)-net over F9, using
(33−8, 33, large)-Net in Base 9 — Upper bound on s
There is no (25, 33, large)-net in base 9, because
- 6 times m-reduction [i] would yield (25, 27, large)-net in base 9, but