Best Known (31, 40, s)-Nets in Base 9
(31, 40, 14761)-Net over F9 — Constructive and digital
Digital (31, 40, 14761)-net over F9, using
- net defined by OOA [i] based on linear OOA(940, 14761, F9, 9, 9) (dual of [(14761, 9), 132809, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(940, 59045, F9, 9) (dual of [59045, 59005, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(940, 59048, F9, 9) (dual of [59048, 59008, 10]-code), using
- 1 times truncation [i] based on linear OA(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times truncation [i] based on linear OA(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(940, 59048, F9, 9) (dual of [59048, 59008, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(940, 59045, F9, 9) (dual of [59045, 59005, 10]-code), using
(31, 40, 59048)-Net over F9 — Digital
Digital (31, 40, 59048)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(940, 59048, F9, 9) (dual of [59048, 59008, 10]-code), using
- 1 times truncation [i] based on linear OA(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times truncation [i] based on linear OA(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using
(31, 40, large)-Net in Base 9 — Upper bound on s
There is no (31, 40, large)-net in base 9, because
- 7 times m-reduction [i] would yield (31, 33, large)-net in base 9, but