Best Known (82, 82+17, s)-Nets in Base 9
(82, 82+17, 132861)-Net over F9 — Constructive and digital
Digital (82, 99, 132861)-net over F9, using
- net defined by OOA [i] based on linear OOA(999, 132861, F9, 17, 17) (dual of [(132861, 17), 2258538, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(999, 1062889, F9, 17) (dual of [1062889, 1062790, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(998, 1062888, F9, 17) (dual of [1062888, 1062790, 18]-code), using
- trace code [i] based on linear OA(8149, 531444, F81, 17) (dual of [531444, 531395, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(8149, 531441, F81, 17) (dual of [531441, 531392, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- trace code [i] based on linear OA(8149, 531444, F81, 17) (dual of [531444, 531395, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(998, 1062888, F9, 17) (dual of [1062888, 1062790, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(999, 1062889, F9, 17) (dual of [1062889, 1062790, 18]-code), using
(82, 82+17, 1062890)-Net over F9 — Digital
Digital (82, 99, 1062890)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(999, 1062890, F9, 17) (dual of [1062890, 1062791, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(998, 1062888, F9, 17) (dual of [1062888, 1062790, 18]-code), using
- trace code [i] based on linear OA(8149, 531444, F81, 17) (dual of [531444, 531395, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(8149, 531441, F81, 17) (dual of [531441, 531392, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- trace code [i] based on linear OA(8149, 531444, F81, 17) (dual of [531444, 531395, 18]-code), using
- linear OA(998, 1062889, F9, 16) (dual of [1062889, 1062791, 17]-code), using Gilbert–Varšamov bound and bm = 998 > Vbs−1(k−1) = 67 161535 276655 786466 344306 903810 165602 895649 648641 161935 720534 620423 756743 127585 718948 043329 [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(998, 1062888, F9, 17) (dual of [1062888, 1062790, 18]-code), using
- construction X with Varšamov bound [i] based on
(82, 82+17, large)-Net in Base 9 — Upper bound on s
There is no (82, 99, large)-net in base 9, because
- 15 times m-reduction [i] would yield (82, 84, large)-net in base 9, but