Best Known (65, 87, s)-Nets in Base 9
(65, 87, 1193)-Net over F9 — Constructive and digital
Digital (65, 87, 1193)-net over F9, using
- 91 times duplication [i] based on digital (64, 86, 1193)-net over F9, using
- net defined by OOA [i] based on linear OOA(986, 1193, F9, 22, 22) (dual of [(1193, 22), 26160, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(986, 13123, F9, 22) (dual of [13123, 13037, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(986, 13126, F9, 22) (dual of [13126, 13040, 23]-code), using
- trace code [i] based on linear OA(8143, 6563, F81, 22) (dual of [6563, 6520, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- trace code [i] based on linear OA(8143, 6563, F81, 22) (dual of [6563, 6520, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(986, 13126, F9, 22) (dual of [13126, 13040, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(986, 13123, F9, 22) (dual of [13123, 13037, 23]-code), using
- net defined by OOA [i] based on linear OOA(986, 1193, F9, 22, 22) (dual of [(1193, 22), 26160, 23]-NRT-code), using
(65, 87, 13128)-Net over F9 — Digital
Digital (65, 87, 13128)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(987, 13128, F9, 22) (dual of [13128, 13041, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(986, 13126, F9, 22) (dual of [13126, 13040, 23]-code), using
- trace code [i] based on linear OA(8143, 6563, F81, 22) (dual of [6563, 6520, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- trace code [i] based on linear OA(8143, 6563, F81, 22) (dual of [6563, 6520, 23]-code), using
- linear OA(986, 13127, F9, 21) (dual of [13127, 13041, 22]-code), using Gilbert–Varšamov bound and bm = 986 > Vbs−1(k−1) = 10767 537645 293308 102918 289574 118116 887974 275199 972694 910181 925261 790755 302354 497009 [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(986, 13126, F9, 22) (dual of [13126, 13040, 23]-code), using
- construction X with Varšamov bound [i] based on
(65, 87, large)-Net in Base 9 — Upper bound on s
There is no (65, 87, large)-net in base 9, because
- 20 times m-reduction [i] would yield (65, 67, large)-net in base 9, but