Best Known (49−12, 49, s)-Nets in Base 9
(49−12, 49, 2188)-Net over F9 — Constructive and digital
Digital (37, 49, 2188)-net over F9, using
- 91 times duplication [i] based on digital (36, 48, 2188)-net over F9, using
- net defined by OOA [i] based on linear OOA(948, 2188, F9, 12, 12) (dual of [(2188, 12), 26208, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(948, 13128, F9, 12) (dual of [13128, 13080, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(948, 13132, F9, 12) (dual of [13132, 13084, 13]-code), using
- trace code [i] based on linear OA(8124, 6566, F81, 12) (dual of [6566, 6542, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [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(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(8124, 6566, F81, 12) (dual of [6566, 6542, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(948, 13132, F9, 12) (dual of [13132, 13084, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(948, 13128, F9, 12) (dual of [13128, 13080, 13]-code), using
- net defined by OOA [i] based on linear OOA(948, 2188, F9, 12, 12) (dual of [(2188, 12), 26208, 13]-NRT-code), using
(49−12, 49, 13134)-Net over F9 — Digital
Digital (37, 49, 13134)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(949, 13134, F9, 12) (dual of [13134, 13085, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(948, 13132, F9, 12) (dual of [13132, 13084, 13]-code), using
- trace code [i] based on linear OA(8124, 6566, F81, 12) (dual of [6566, 6542, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [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(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(8124, 6566, F81, 12) (dual of [6566, 6542, 13]-code), using
- linear OA(948, 13133, F9, 11) (dual of [13133, 13085, 12]-code), using Gilbert–Varšamov bound and bm = 948 > Vbs−1(k−1) = 44 977816 878330 409637 265711 238376 073427 167969 [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(948, 13132, F9, 12) (dual of [13132, 13084, 13]-code), using
- construction X with Varšamov bound [i] based on
(49−12, 49, large)-Net in Base 9 — Upper bound on s
There is no (37, 49, large)-net in base 9, because
- 10 times m-reduction [i] would yield (37, 39, large)-net in base 9, but