Best Known (136, 136+13, s)-Nets in Base 9
(136, 136+13, 5592767)-Net over F9 — Constructive and digital
Digital (136, 149, 5592767)-net over F9, using
- generalized (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 (35, 41, 2796200)-net over F9, using
- s-reduction based on digital (35, 41, 2796201)-net over F9, using
- net defined by OOA [i] based on linear OOA(941, 2796201, F9, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(941, large, F9, 6) (dual of [large, large−41, 7]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(941, large, F9, 6) (dual of [large, large−41, 7]-code), using
- net defined by OOA [i] based on linear OOA(941, 2796201, F9, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- s-reduction based on digital (35, 41, 2796201)-net over F9, using
- digital (85, 98, 2796200)-net over F9, using
- net defined by OOA [i] based on linear OOA(998, 2796200, F9, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(998, 8388601, F9, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(998, 8388602, F9, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- trace code [i] based on 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
- trace code [i] based on linear OOA(8149, 4194301, F81, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(998, 8388602, F9, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(998, 8388601, F9, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(998, 2796200, F9, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- digital (6, 10, 367)-net over F9, using
(136, 136+13, large)-Net over F9 — Digital
Digital (136, 149, large)-net over F9, using
- 94 times duplication [i] based on digital (132, 145, large)-net over F9, using
- t-expansion [i] based on digital (124, 145, large)-net over F9, using
(136, 136+13, large)-Net in Base 9 — Upper bound on s
There is no (136, 149, large)-net in base 9, because
- 11 times m-reduction [i] would yield (136, 138, large)-net in base 9, but