Best Known (83−18, 83, s)-Nets in Base 9
(83−18, 83, 6562)-Net over F9 — Constructive and digital
Digital (65, 83, 6562)-net over F9, using
- t-expansion [i] based on digital (64, 83, 6562)-net over F9, using
- net defined by OOA [i] based on linear OOA(983, 6562, F9, 19, 19) (dual of [(6562, 19), 124595, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(983, 59059, F9, 19) (dual of [59059, 58976, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(981, 59049, F9, 19) (dual of [59049, 58968, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(971, 59049, F9, 16) (dual of [59049, 58978, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(18) ⊂ Ce(15) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(983, 59059, F9, 19) (dual of [59059, 58976, 20]-code), using
- net defined by OOA [i] based on linear OOA(983, 6562, F9, 19, 19) (dual of [(6562, 19), 124595, 20]-NRT-code), using
(83−18, 83, 59062)-Net over F9 — Digital
Digital (65, 83, 59062)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(983, 59062, F9, 18) (dual of [59062, 58979, 19]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- linear OA(981, 59049, F9, 19) (dual of [59049, 58968, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(971, 59049, F9, 16) (dual of [59049, 58978, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(966, 59049, F9, 15) (dual of [59049, 58983, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(91, 12, F9, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(14) [i] based on
(83−18, 83, large)-Net in Base 9 — Upper bound on s
There is no (65, 83, large)-net in base 9, because
- 16 times m-reduction [i] would yield (65, 67, large)-net in base 9, but