Best Known (136, 136+14, s)-Nets in Base 9
(136, 136+14, 3991070)-Net over F9 — Constructive and digital
Digital (136, 150, 3991070)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (37, 44, 1594328)-net over F9, using
- net defined by OOA [i] based on linear OOA(944, 1594328, F9, 7, 7) (dual of [(1594328, 7), 11160252, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(944, 4782985, F9, 7) (dual of [4782985, 4782941, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(943, 4782970, F9, 7) (dual of [4782970, 4782927, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(929, 4782970, F9, 5) (dual of [4782970, 4782941, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(91, 15, F9, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(944, 4782985, F9, 7) (dual of [4782985, 4782941, 8]-code), using
- net defined by OOA [i] based on linear OOA(944, 1594328, F9, 7, 7) (dual of [(1594328, 7), 11160252, 8]-NRT-code), using
- digital (92, 106, 2396742)-net over F9, using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F81, using
- net defined by OOA [i] based on linear OOA(8153, 1198371, F81, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(8153, 8388597, F81, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(8153, 8388597, F81, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(8153, 1198371, F81, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F81, using
- digital (37, 44, 1594328)-net over F9, using
(136, 136+14, large)-Net over F9 — Digital
Digital (136, 150, large)-net over F9, using
- 95 times duplication [i] based on digital (131, 145, large)-net over F9, using
- t-expansion [i] based on digital (124, 145, large)-net over F9, using
(136, 136+14, large)-Net in Base 9 — Upper bound on s
There is no (136, 150, large)-net in base 9, because
- 12 times m-reduction [i] would yield (136, 138, large)-net in base 9, but