Best Known (74, 74+25, s)-Nets in Base 9
(74, 74+25, 1093)-Net over F9 — Constructive and digital
Digital (74, 99, 1093)-net over F9, using
- 91 times duplication [i] based on digital (73, 98, 1093)-net over F9, using
- net defined by OOA [i] based on linear OOA(998, 1093, F9, 25, 25) (dual of [(1093, 25), 27227, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(998, 13117, F9, 25) (dual of [13117, 13019, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(998, 13124, F9, 25) (dual of [13124, 13026, 26]-code), using
- trace code [i] based on linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- trace code [i] based on linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(998, 13124, F9, 25) (dual of [13124, 13026, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(998, 13117, F9, 25) (dual of [13117, 13019, 26]-code), using
- net defined by OOA [i] based on linear OOA(998, 1093, F9, 25, 25) (dual of [(1093, 25), 27227, 26]-NRT-code), using
(74, 74+25, 13128)-Net over F9 — Digital
Digital (74, 99, 13128)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(999, 13128, F9, 25) (dual of [13128, 13029, 26]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(998, 13126, F9, 25) (dual of [13126, 13028, 26]-code), using
- trace code [i] based on linear OA(8149, 6563, F81, 25) (dual of [6563, 6514, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(8149, 6561, F81, 25) (dual of [6561, 6512, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(24) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(8149, 6563, F81, 25) (dual of [6563, 6514, 26]-code), using
- linear OA(998, 13127, F9, 24) (dual of [13127, 13029, 25]-code), using Gilbert–Varšamov bound and bm = 998 > Vbs−1(k−1) = 1167 723636 908507 840791 508470 961814 717195 894593 908268 727091 005488 371972 029102 239022 728293 953009 [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(998, 13126, F9, 25) (dual of [13126, 13028, 26]-code), using
- construction X with Varšamov bound [i] based on
(74, 74+25, large)-Net in Base 9 — Upper bound on s
There is no (74, 99, large)-net in base 9, because
- 23 times m-reduction [i] would yield (74, 76, large)-net in base 9, but