Best Known (37, 37+8, s)-Nets in Base 9
(37, 37+8, 265722)-Net over F9 — Constructive and digital
Digital (37, 45, 265722)-net over F9, using
- 91 times duplication [i] based on digital (36, 44, 265722)-net over F9, using
- net defined by OOA [i] based on linear OOA(944, 265722, F9, 8, 8) (dual of [(265722, 8), 2125732, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(944, 1062888, F9, 8) (dual of [1062888, 1062844, 9]-code), using
- trace code [i] based on linear OA(8122, 531444, F81, 8) (dual of [531444, 531422, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(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(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(7) ⊂ Ce(6) [i] based on
- trace code [i] based on linear OA(8122, 531444, F81, 8) (dual of [531444, 531422, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(944, 1062888, F9, 8) (dual of [1062888, 1062844, 9]-code), using
- net defined by OOA [i] based on linear OOA(944, 265722, F9, 8, 8) (dual of [(265722, 8), 2125732, 9]-NRT-code), using
(37, 37+8, 1062890)-Net over F9 — Digital
Digital (37, 45, 1062890)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(945, 1062890, F9, 8) (dual of [1062890, 1062845, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(944, 1062888, F9, 8) (dual of [1062888, 1062844, 9]-code), using
- trace code [i] based on linear OA(8122, 531444, F81, 8) (dual of [531444, 531422, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(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(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(7) ⊂ Ce(6) [i] based on
- trace code [i] based on linear OA(8122, 531444, F81, 8) (dual of [531444, 531422, 9]-code), using
- linear OA(944, 1062889, F9, 7) (dual of [1062889, 1062845, 8]-code), using Gilbert–Varšamov bound and bm = 944 > Vbs−1(k−1) = 524 960499 617466 497496 172048 336556 890689 [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(944, 1062888, F9, 8) (dual of [1062888, 1062844, 9]-code), using
- construction X with Varšamov bound [i] based on
(37, 37+8, large)-Net in Base 9 — Upper bound on s
There is no (37, 45, large)-net in base 9, because
- 6 times m-reduction [i] would yield (37, 39, large)-net in base 9, but