Best Known (135, 147, s)-Nets in Base 9
(135, 147, 5621927)-Net over F9 — Constructive and digital
Digital (135, 147, 5621927)-net over F9, using
- generalized (u, u+v)-construction [i] based on
- digital (12, 16, 29527)-net over F9, using
- net defined by OOA [i] based on linear OOA(916, 29527, F9, 4, 4) (dual of [(29527, 4), 118092, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(916, 29527, F9, 3, 4) (dual of [(29527, 3), 88565, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(916, 59054, F9, 4) (dual of [59054, 59038, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(911, 59049, F9, 3) (dual of [59049, 59038, 4]-code or 59049-cap in PG(10,9)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 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
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(916, 59054, F9, 4) (dual of [59054, 59038, 5]-code), using
- appending kth column [i] based on linear OOA(916, 29527, F9, 3, 4) (dual of [(29527, 3), 88565, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(916, 29527, F9, 4, 4) (dual of [(29527, 4), 118092, 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 (78, 90, 2796200)-net over F9, using
- net defined by OOA [i] based on linear OOA(990, 2796200, F9, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(990, 8388601, F9, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(990, 8388602, F9, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- trace code [i] based on 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
- trace code [i] based on linear OOA(8145, 4194301, F81, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(990, 8388602, F9, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(990, 8388601, F9, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(990, 2796200, F9, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- digital (12, 16, 29527)-net over F9, using
(135, 147, large)-Net over F9 — Digital
Digital (135, 147, large)-net over F9, using
- 92 times duplication [i] based on digital (133, 145, large)-net over F9, using
- t-expansion [i] based on digital (124, 145, large)-net over F9, using
(135, 147, large)-Net in Base 9 — Upper bound on s
There is no (135, 147, large)-net in base 9, because
- 10 times m-reduction [i] would yield (135, 137, large)-net in base 9, but