Best Known (68−8, 68, s)-Nets in Base 9
(68−8, 68, 4194667)-Net over F9 — Constructive and digital
Digital (60, 68, 4194667)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (6, 10, 367)-net over F9, using
- net defined by OOA [i] based on linear OOA(910, 367, F9, 4, 4) (dual of [(367, 4), 1458, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(910, 367, F9, 3, 4) (dual of [(367, 3), 1091, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(910, 734, F9, 4) (dual of [734, 724, 5]-code), using
- construction XX applied to C1 = C([727,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([727,2]) [i] based on
- linear OA(97, 728, F9, 3) (dual of [728, 721, 4]-code or 728-cap in PG(6,9)), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(97, 728, F9, 3) (dual of [728, 721, 4]-code or 728-cap in PG(6,9)), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(910, 728, F9, 4) (dual of [728, 718, 5]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(94, 728, F9, 2) (dual of [728, 724, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(90, 3, F9, 0) (dual of [3, 3, 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(90, 3, F9, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([727,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([727,2]) [i] based on
- OA 2-folding and stacking [i] based on linear OA(910, 734, F9, 4) (dual of [734, 724, 5]-code), using
- appending kth column [i] based on linear OOA(910, 367, F9, 3, 4) (dual of [(367, 3), 1091, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(910, 367, F9, 4, 4) (dual of [(367, 4), 1458, 5]-NRT-code), using
- digital (50, 58, 4194300)-net over F9, using
- net defined by OOA [i] based on linear OOA(958, 4194300, F9, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(958, 8388601, F9, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(958, 8388602, F9, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- trace code [i] based on linear OOA(8129, 4194301, F81, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8129, 8388602, F81, 8) (dual of [8388602, 8388573, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- OOA 2-folding [i] based on linear OA(8129, 8388602, F81, 8) (dual of [8388602, 8388573, 9]-code), using
- trace code [i] based on linear OOA(8129, 4194301, F81, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(958, 8388602, F9, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(958, 8388601, F9, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(958, 4194300, F9, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- digital (6, 10, 367)-net over F9, using
(68−8, 68, large)-Net over F9 — Digital
Digital (60, 68, large)-net over F9, using
- 93 times duplication [i] based on digital (57, 65, large)-net over F9, using
- t-expansion [i] based on digital (55, 65, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(965, large, F9, 10) (dual of [large, large−65, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(965, large, F9, 10) (dual of [large, large−65, 11]-code), using
- t-expansion [i] based on digital (55, 65, large)-net over F9, using
(68−8, 68, large)-Net in Base 9 — Upper bound on s
There is no (60, 68, large)-net in base 9, because
- 6 times m-reduction [i] would yield (60, 62, large)-net in base 9, but