Best Known (47−12, 47, s)-Nets in Base 9
(47−12, 47, 2187)-Net over F9 — Constructive and digital
Digital (35, 47, 2187)-net over F9, using
- 91 times duplication [i] based on digital (34, 46, 2187)-net over F9, using
- net defined by OOA [i] based on linear OOA(946, 2187, F9, 12, 12) (dual of [(2187, 12), 26198, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(946, 13122, F9, 12) (dual of [13122, 13076, 13]-code), using
- trace code [i] based on linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- trace code [i] based on linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(946, 13122, F9, 12) (dual of [13122, 13076, 13]-code), using
- net defined by OOA [i] based on linear OOA(946, 2187, F9, 12, 12) (dual of [(2187, 12), 26198, 13]-NRT-code), using
(47−12, 47, 13128)-Net over F9 — Digital
Digital (35, 47, 13128)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(947, 13128, F9, 12) (dual of [13128, 13081, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(946, 13126, F9, 12) (dual of [13126, 13080, 13]-code), using
- trace code [i] based on linear OA(8123, 6563, F81, 12) (dual of [6563, 6540, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 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(11) ⊂ Ce(10) [i] based on
- trace code [i] based on linear OA(8123, 6563, F81, 12) (dual of [6563, 6540, 13]-code), using
- linear OA(946, 13127, F9, 11) (dual of [13127, 13081, 12]-code), using Gilbert–Varšamov bound and bm = 946 > Vbs−1(k−1) = 44 772667 398243 116695 968095 004425 436474 945009 [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(946, 13126, F9, 12) (dual of [13126, 13080, 13]-code), using
- construction X with Varšamov bound [i] based on
(47−12, 47, large)-Net in Base 9 — Upper bound on s
There is no (35, 47, large)-net in base 9, because
- 10 times m-reduction [i] would yield (35, 37, large)-net in base 9, but