Best Known (83, 83+9, s)-Nets in Base 9
(83, 83+9, 8388600)-Net over F9 — Constructive and digital
Digital (83, 92, 8388600)-net over F9, using
- trace code for nets [i] based on digital (37, 46, 4194300)-net over F81, using
- net defined by OOA [i] based on linear OOA(8146, 4194300, F81, 10, 9) (dual of [(4194300, 10), 41942954, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8146, 8388601, F81, 2, 9) (dual of [(8388601, 2), 16777156, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 8388602, F81, 2, 9) (dual of [(8388602, 2), 16777158, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(8113, 4194301, F81, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8113, 8388602, F81, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- OOA 2-folding [i] based on linear OA(8113, 8388602, F81, 4) (dual of [8388602, 8388589, 5]-code), using
- 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
- linear OOA(8113, 4194301, F81, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(8146, 8388602, F81, 2, 9) (dual of [(8388602, 2), 16777158, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8146, 8388601, F81, 2, 9) (dual of [(8388601, 2), 16777156, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8146, 4194300, F81, 10, 9) (dual of [(4194300, 10), 41942954, 10]-NRT-code), using
(83, 83+9, 8388602)-Net in Base 9 — Constructive
(83, 92, 8388602)-net in base 9, using
- (u, u+v)-construction [i] based on
- (20, 24, 4194301)-net in base 9, using
- base change [i] based on digital (12, 16, 4194301)-net over F27, using
- net defined by OOA [i] based on linear OOA(2716, 4194301, F27, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2716, 8388602, F27, 4) (dual of [8388602, 8388586, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(2716, large, F27, 4) (dual of [large, large−16, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(2716, large, F27, 4) (dual of [large, large−16, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(2716, 8388602, F27, 4) (dual of [8388602, 8388586, 5]-code), using
- net defined by OOA [i] based on linear OOA(2716, 4194301, F27, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
- base change [i] based on digital (12, 16, 4194301)-net over F27, using
- digital (59, 68, 4194301)-net over F9, using
- net defined by OOA [i] based on linear OOA(968, 4194301, F9, 10, 9) (dual of [(4194301, 10), 41942942, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(968, large, F9, 2, 9), using
- 1 times NRT-code embedding in larger space [i] 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
- 1 times NRT-code embedding in larger space [i] 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(968, large, F9, 2, 9), using
- net defined by OOA [i] based on linear OOA(968, 4194301, F9, 10, 9) (dual of [(4194301, 10), 41942942, 10]-NRT-code), using
- (20, 24, 4194301)-net in base 9, using
(83, 83+9, large)-Net over F9 — Digital
Digital (83, 92, large)-net over F9, using
- 5 times m-reduction [i] based on digital (83, 97, large)-net over F9, using
(83, 83+9, large)-Net in Base 9 — Upper bound on s
There is no (83, 92, large)-net in base 9, because
- 7 times m-reduction [i] would yield (83, 85, large)-net in base 9, but