Best Known (33−9, 33, s)-Nets in Base 9
(33−9, 33, 1641)-Net over F9 — Constructive and digital
Digital (24, 33, 1641)-net over F9, using
- net defined by OOA [i] based on linear OOA(933, 1641, F9, 9, 9) (dual of [(1641, 9), 14736, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(933, 6565, F9, 9) (dual of [6565, 6532, 10]-code), using
- 1 times truncation [i] based on linear OA(934, 6566, F9, 10) (dual of [6566, 6532, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(929, 6561, F9, 8) (dual of [6561, 6532, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(91, 5, F9, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(934, 6566, F9, 10) (dual of [6566, 6532, 11]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(933, 6565, F9, 9) (dual of [6565, 6532, 10]-code), using
(33−9, 33, 6565)-Net over F9 — Digital
Digital (24, 33, 6565)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(933, 6565, F9, 9) (dual of [6565, 6532, 10]-code), using
- 1 times truncation [i] based on linear OA(934, 6566, F9, 10) (dual of [6566, 6532, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(929, 6561, F9, 8) (dual of [6561, 6532, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(91, 5, F9, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(934, 6566, F9, 10) (dual of [6566, 6532, 11]-code), using
(33−9, 33, large)-Net in Base 9 — Upper bound on s
There is no (24, 33, large)-net in base 9, because
- 7 times m-reduction [i] would yield (24, 26, large)-net in base 9, but