Best Known (50−9, 50, s)-Nets in Base 9
(50−9, 50, 265721)-Net over F9 — Constructive and digital
Digital (41, 50, 265721)-net over F9, using
- net defined by OOA [i] based on linear OOA(950, 265721, F9, 9, 9) (dual of [(265721, 9), 2391439, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(950, 1062885, F9, 9) (dual of [1062885, 1062835, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(950, 1062888, F9, 9) (dual of [1062888, 1062838, 10]-code), using
- trace code [i] based on linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(8125, 531441, F81, 9) (dual of [531441, 531416, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- trace code [i] based on linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(950, 1062888, F9, 9) (dual of [1062888, 1062838, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(950, 1062885, F9, 9) (dual of [1062885, 1062835, 10]-code), using
(50−9, 50, 1062888)-Net over F9 — Digital
Digital (41, 50, 1062888)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(950, 1062888, F9, 9) (dual of [1062888, 1062838, 10]-code), using
- trace code [i] based on linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(8125, 531441, F81, 9) (dual of [531441, 531416, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- trace code [i] based on linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
(50−9, 50, large)-Net in Base 9 — Upper bound on s
There is no (41, 50, large)-net in base 9, because
- 7 times m-reduction [i] would yield (41, 43, large)-net in base 9, but