Best Known (121, 121+12, s)-Nets in Base 9
(121, 121+12, 5592400)-Net over F9 — Constructive and digital
Digital (121, 133, 5592400)-net over F9, using
- 91 times duplication [i] based on digital (120, 132, 5592400)-net over F9, using
- trace code for nets [i] based on digital (54, 66, 2796200)-net over F81, using
- net defined by OOA [i] based on linear OOA(8166, 2796200, F81, 14, 12) (dual of [(2796200, 14), 39146734, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8166, 8388601, F81, 2, 12) (dual of [(8388601, 2), 16777136, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8166, 8388602, F81, 2, 12) (dual of [(8388602, 2), 16777138, 13]-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(8145, 4194301, F81, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8145, 8388602, F81, 12) (dual of [8388602, 8388557, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- OOA 2-folding [i] based on linear OA(8145, 8388602, F81, 12) (dual of [8388602, 8388557, 13]-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(8166, 8388602, F81, 2, 12) (dual of [(8388602, 2), 16777138, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8166, 8388601, F81, 2, 12) (dual of [(8388601, 2), 16777136, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8166, 2796200, F81, 14, 12) (dual of [(2796200, 14), 39146734, 13]-NRT-code), using
- trace code for nets [i] based on digital (54, 66, 2796200)-net over F81, using
(121, 121+12, 5592402)-Net in Base 9 — Constructive
(121, 133, 5592402)-net in base 9, using
- 91 times duplication [i] based on (120, 132, 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 (81, 93, 2796201)-net over F9, using
- net defined by OOA [i] based on linear OOA(993, 2796201, F9, 15, 12) (dual of [(2796201, 15), 41942922, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(993, 5592403, F9, 3, 12) (dual of [(5592403, 3), 16777116, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(990, 5592402, F9, 3, 12) (dual of [(5592402, 3), 16777116, 13]-NRT-code), using
- trace code [i] based on linear OOA(8145, 2796201, F81, 3, 12) (dual of [(2796201, 3), 8388558, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 3-folding [i] based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- trace code [i] based on linear OOA(8145, 2796201, F81, 3, 12) (dual of [(2796201, 3), 8388558, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(990, 5592402, F9, 3, 12) (dual of [(5592402, 3), 16777116, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(993, 5592403, F9, 3, 12) (dual of [(5592403, 3), 16777116, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(993, 2796201, F9, 15, 12) (dual of [(2796201, 15), 41942922, 13]-NRT-code), using
- (33, 39, 2796201)-net in base 9, using
- (u, u+v)-construction [i] based on
(121, 121+12, large)-Net over F9 — Digital
Digital (121, 133, large)-net over F9, using
- t-expansion [i] based on digital (117, 133, large)-net over F9, using
- 4 times m-reduction [i] based on digital (117, 137, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(9137, large, F9, 20) (dual of [large, large−137, 21]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(9137, large, F9, 20) (dual of [large, large−137, 21]-code), using
- 4 times m-reduction [i] based on digital (117, 137, large)-net over F9, using
(121, 121+12, large)-Net in Base 9 — Upper bound on s
There is no (121, 133, large)-net in base 9, because
- 10 times m-reduction [i] would yield (121, 123, large)-net in base 9, but