Best Known (71−18, 71, s)-Nets in Base 9
(71−18, 71, 1458)-Net over F9 — Constructive and digital
Digital (53, 71, 1458)-net over F9, using
- 91 times duplication [i] based on digital (52, 70, 1458)-net over F9, using
- net defined by OOA [i] based on linear OOA(970, 1458, F9, 18, 18) (dual of [(1458, 18), 26174, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(970, 13122, F9, 18) (dual of [13122, 13052, 19]-code), using
- trace code [i] based on linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- trace code [i] based on linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(970, 13122, F9, 18) (dual of [13122, 13052, 19]-code), using
- net defined by OOA [i] based on linear OOA(970, 1458, F9, 18, 18) (dual of [(1458, 18), 26174, 19]-NRT-code), using
(71−18, 71, 12704)-Net over F9 — Digital
Digital (53, 71, 12704)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(971, 12704, F9, 18) (dual of [12704, 12633, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(971, 13127, F9, 18) (dual of [13127, 13056, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(970, 13126, F9, 18) (dual of [13126, 13056, 19]-code), using
- trace code [i] based on linear OA(8135, 6563, F81, 18) (dual of [6563, 6528, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- trace code [i] based on linear OA(8135, 6563, F81, 18) (dual of [6563, 6528, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(970, 13126, F9, 18) (dual of [13126, 13056, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(971, 13127, F9, 18) (dual of [13127, 13056, 19]-code), using
(71−18, 71, large)-Net in Base 9 — Upper bound on s
There is no (53, 71, large)-net in base 9, because
- 16 times m-reduction [i] would yield (53, 55, large)-net in base 9, but