Best Known (40−11, 40, s)-Nets in Base 9
(40−11, 40, 1314)-Net over F9 — Constructive and digital
Digital (29, 40, 1314)-net over F9, using
- 91 times duplication [i] based on digital (28, 39, 1314)-net over F9, using
- net defined by OOA [i] based on linear OOA(939, 1314, F9, 11, 11) (dual of [(1314, 11), 14415, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(939, 6571, F9, 11) (dual of [6571, 6532, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(937, 6561, F9, 11) (dual of [6561, 6524, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(92, 10, F9, 2) (dual of [10, 8, 3]-code or 10-arc in PG(1,9)), using
- extended Reed–Solomon code RSe(8,9) [i]
- Hamming code H(2,9) [i]
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(939, 6571, F9, 11) (dual of [6571, 6532, 12]-code), using
- net defined by OOA [i] based on linear OOA(939, 1314, F9, 11, 11) (dual of [(1314, 11), 14415, 12]-NRT-code), using
(40−11, 40, 6573)-Net over F9 — Digital
Digital (29, 40, 6573)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(940, 6573, F9, 11) (dual of [6573, 6533, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(939, 6571, F9, 11) (dual of [6571, 6532, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(937, 6561, F9, 11) (dual of [6561, 6524, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(92, 10, F9, 2) (dual of [10, 8, 3]-code or 10-arc in PG(1,9)), using
- extended Reed–Solomon code RSe(8,9) [i]
- Hamming code H(2,9) [i]
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(939, 6572, F9, 10) (dual of [6572, 6533, 11]-code), using Gilbert–Varšamov bound and bm = 939 > Vbs−1(k−1) = 8 402802 273975 511516 004704 235432 749337 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(939, 6571, F9, 11) (dual of [6571, 6532, 12]-code), using
- construction X with Varšamov bound [i] based on
(40−11, 40, large)-Net in Base 9 — Upper bound on s
There is no (29, 40, large)-net in base 9, because
- 9 times m-reduction [i] would yield (29, 31, large)-net in base 9, but