Best Known (39−7, 39, s)-Nets in Base 9
(39−7, 39, 354296)-Net over F9 — Constructive and digital
Digital (32, 39, 354296)-net over F9, using
- net defined by OOA [i] based on linear OOA(939, 354296, F9, 7, 7) (dual of [(354296, 7), 2480033, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(939, 1062889, F9, 7) (dual of [1062889, 1062850, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(938, 1062888, F9, 7) (dual of [1062888, 1062850, 8]-code), using
- trace code [i] based on linear OA(8119, 531444, F81, 7) (dual of [531444, 531425, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(8119, 531441, F81, 7) (dual of [531441, 531422, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(8119, 531444, F81, 7) (dual of [531444, 531425, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(938, 1062888, F9, 7) (dual of [1062888, 1062850, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(939, 1062889, F9, 7) (dual of [1062889, 1062850, 8]-code), using
(39−7, 39, 1062890)-Net over F9 — Digital
Digital (32, 39, 1062890)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(939, 1062890, F9, 7) (dual of [1062890, 1062851, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(938, 1062888, F9, 7) (dual of [1062888, 1062850, 8]-code), using
- trace code [i] based on linear OA(8119, 531444, F81, 7) (dual of [531444, 531425, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(8119, 531441, F81, 7) (dual of [531441, 531422, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(8119, 531444, F81, 7) (dual of [531444, 531425, 8]-code), using
- linear OA(938, 1062889, F9, 6) (dual of [1062889, 1062851, 7]-code), using Gilbert–Varšamov bound and bm = 938 > Vbs−1(k−1) = 370 426781 131608 262412 300373 957185 [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(938, 1062888, F9, 7) (dual of [1062888, 1062850, 8]-code), using
- construction X with Varšamov bound [i] based on
(39−7, 39, large)-Net in Base 9 — Upper bound on s
There is no (32, 39, large)-net in base 9, because
- 5 times m-reduction [i] would yield (32, 34, large)-net in base 9, but