Best Known (82−18, 82, s)-Nets in Base 9
(82−18, 82, 6562)-Net over F9 — Constructive and digital
Digital (64, 82, 6562)-net over F9, using
- net defined by OOA [i] based on linear OOA(982, 6562, F9, 18, 18) (dual of [(6562, 18), 118034, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(982, 59058, F9, 18) (dual of [59058, 58976, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(982, 59060, F9, 18) (dual of [59060, 58978, 19]-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(91, 11, F9, 1) (dual of [11, 10, 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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(982, 59060, F9, 18) (dual of [59060, 58978, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(982, 59058, F9, 18) (dual of [59058, 58976, 19]-code), using
(82−18, 82, 57575)-Net over F9 — Digital
Digital (64, 82, 57575)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(982, 57575, F9, 18) (dual of [57575, 57493, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(982, 59060, F9, 18) (dual of [59060, 58978, 19]-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(91, 11, F9, 1) (dual of [11, 10, 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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(982, 59060, F9, 18) (dual of [59060, 58978, 19]-code), using
(82−18, 82, large)-Net in Base 9 — Upper bound on s
There is no (64, 82, large)-net in base 9, because
- 16 times m-reduction [i] would yield (64, 66, large)-net in base 9, but