Best Known (74, 83, s)-Nets in Base 9
(74, 83, 4223830)-Net over F9 — Constructive and digital
Digital (74, 83, 4223830)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (13, 17, 29530)-net over F9, using
- net defined by OOA [i] based on linear OOA(917, 29530, F9, 4, 4) (dual of [(29530, 4), 118103, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(917, 29530, F9, 3, 4) (dual of [(29530, 3), 88573, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(917, 59060, F9, 4) (dual of [59060, 59043, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(96, 59049, F9, 2) (dual of [59049, 59043, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(91, 11, F9, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(917, 59060, F9, 4) (dual of [59060, 59043, 5]-code), using
- appending kth column [i] based on linear OOA(917, 29530, F9, 3, 4) (dual of [(29530, 3), 88573, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(917, 29530, F9, 4, 4) (dual of [(29530, 4), 118103, 5]-NRT-code), using
- digital (57, 66, 4194300)-net over F9, using
- net defined by OOA [i] based on linear OOA(966, 4194300, F9, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(966, 8388601, F9, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(966, 8388602, F9, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
- trace code [i] based on linear OOA(8133, 4194301, F81, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8133, 8388602, F81, 9) (dual of [8388602, 8388569, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- OOA 2-folding [i] based on linear OA(8133, 8388602, F81, 9) (dual of [8388602, 8388569, 10]-code), using
- trace code [i] based on linear OOA(8133, 4194301, F81, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(966, 8388602, F9, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(966, 8388601, F9, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(966, 4194300, F9, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- digital (13, 17, 29530)-net over F9, using
(74, 83, large)-Net over F9 — Digital
Digital (74, 83, large)-net over F9, using
- 92 times duplication [i] based on digital (72, 81, large)-net over F9, using
- t-expansion [i] based on digital (69, 81, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(981, large, F9, 12) (dual of [large, large−81, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(981, large, F9, 12) (dual of [large, large−81, 13]-code), using
- t-expansion [i] based on digital (69, 81, large)-net over F9, using
(74, 83, large)-Net in Base 9 — Upper bound on s
There is no (74, 83, large)-net in base 9, because
- 7 times m-reduction [i] would yield (74, 76, large)-net in base 9, but