Best Known (140−13, 140, s)-Nets in Base 9
(140−13, 140, 5592400)-Net over F9 — Constructive and digital
Digital (127, 140, 5592400)-net over F9, using
- trace code for nets [i] based on digital (57, 70, 2796200)-net over F81, using
- net defined by OOA [i] based on linear OOA(8170, 2796200, F81, 14, 13) (dual of [(2796200, 14), 39146730, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8170, 8388601, F81, 2, 13) (dual of [(8388601, 2), 16777132, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8170, 8388602, F81, 2, 13) (dual of [(8388602, 2), 16777134, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(8121, 4194301, F81, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8121, 8388602, F81, 6) (dual of [8388602, 8388581, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- OOA 2-folding [i] based on linear OA(8121, 8388602, F81, 6) (dual of [8388602, 8388581, 7]-code), using
- linear OOA(8149, 4194301, F81, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8149, 8388602, F81, 13) (dual of [8388602, 8388553, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- OOA 2-folding [i] based on linear OA(8149, 8388602, F81, 13) (dual of [8388602, 8388553, 14]-code), using
- linear OOA(8121, 4194301, F81, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(8170, 8388602, F81, 2, 13) (dual of [(8388602, 2), 16777134, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8170, 8388601, F81, 2, 13) (dual of [(8388601, 2), 16777132, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8170, 2796200, F81, 14, 13) (dual of [(2796200, 14), 39146730, 14]-NRT-code), using
(140−13, 140, 5592402)-Net in Base 9 — Constructive
(127, 140, 5592402)-net in base 9, using
- (u, u+v)-construction [i] based on
- (33, 39, 2796201)-net in base 9, using
- base change [i] based on digital (20, 26, 2796201)-net over F27, using
- net defined by OOA [i] based on linear OOA(2726, 2796201, F27, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2726, large, F27, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(2726, large, F27, 6) (dual of [large, large−26, 7]-code), using
- net defined by OOA [i] based on linear OOA(2726, 2796201, F27, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- base change [i] based on digital (20, 26, 2796201)-net over F27, using
- digital (88, 101, 2796201)-net over F9, using
- net defined by OOA [i] based on linear OOA(9101, 2796201, F9, 15, 13) (dual of [(2796201, 15), 41942914, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(9101, 5592403, F9, 3, 13) (dual of [(5592403, 3), 16777108, 14]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(998, 5592402, F9, 3, 13) (dual of [(5592402, 3), 16777108, 14]-NRT-code), using
- trace code [i] based on linear OOA(8149, 2796201, F81, 3, 13) (dual of [(2796201, 3), 8388554, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- trace code [i] based on linear OOA(8149, 2796201, F81, 3, 13) (dual of [(2796201, 3), 8388554, 14]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(998, 5592402, F9, 3, 13) (dual of [(5592402, 3), 16777108, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(9101, 5592403, F9, 3, 13) (dual of [(5592403, 3), 16777108, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(9101, 2796201, F9, 15, 13) (dual of [(2796201, 15), 41942914, 14]-NRT-code), using
- (33, 39, 2796201)-net in base 9, using
(140−13, 140, large)-Net over F9 — Digital
Digital (127, 140, large)-net over F9, using
- t-expansion [i] based on digital (124, 140, large)-net over F9, using
- 5 times m-reduction [i] based on digital (124, 145, large)-net over F9, using
(140−13, 140, large)-Net in Base 9 — Upper bound on s
There is no (127, 140, large)-net in base 9, because
- 11 times m-reduction [i] would yield (127, 129, large)-net in base 9, but