Best Known (67, 89, s)-Nets in Base 9
(67, 89, 1193)-Net over F9 — Constructive and digital
Digital (67, 89, 1193)-net over F9, using
- 1 times m-reduction [i] based on digital (67, 90, 1193)-net over F9, using
- net defined by OOA [i] based on linear OOA(990, 1193, F9, 23, 23) (dual of [(1193, 23), 27349, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(990, 13124, F9, 23) (dual of [13124, 13034, 24]-code), using
- trace code [i] based on linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- trace code [i] based on linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(990, 13124, F9, 23) (dual of [13124, 13034, 24]-code), using
- net defined by OOA [i] based on linear OOA(990, 1193, F9, 23, 23) (dual of [(1193, 23), 27349, 24]-NRT-code), using
(67, 89, 13134)-Net over F9 — Digital
Digital (67, 89, 13134)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(989, 13134, F9, 22) (dual of [13134, 13045, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(988, 13132, F9, 22) (dual of [13132, 13044, 23]-code), using
- trace code [i] based on linear OA(8144, 6566, F81, 22) (dual of [6566, 6522, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [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(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(21) ⊂ Ce(19) [i] based on
- trace code [i] based on linear OA(8144, 6566, F81, 22) (dual of [6566, 6522, 23]-code), using
- linear OA(988, 13133, F9, 21) (dual of [13133, 13045, 22]-code), using Gilbert–Varšamov bound and bm = 988 > Vbs−1(k−1) = 10866 475840 009129 446712 231769 625490 802448 112523 821323 343049 049806 412212 865038 559969 [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(988, 13132, F9, 22) (dual of [13132, 13044, 23]-code), using
- construction X with Varšamov bound [i] based on
(67, 89, large)-Net in Base 9 — Upper bound on s
There is no (67, 89, large)-net in base 9, because
- 20 times m-reduction [i] would yield (67, 69, large)-net in base 9, but