Best Known (150−12, 150, s)-Nets in Base 9
(150−12, 150, 5984856)-Net over F9 — Constructive and digital
Digital (138, 150, 5984856)-net over F9, using
- 91 times duplication [i] based on digital (137, 149, 5984856)-net over F9, using
- generalized (u, u+v)-construction [i] based on
- digital (18, 22, 1594328)-net over F9, using
- s-reduction based on digital (18, 22, 2391488)-net over F9, using
- net defined by OOA [i] based on linear OOA(922, 2391488, F9, 4, 4) (dual of [(2391488, 4), 9565930, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(922, 2391488, F9, 3, 4) (dual of [(2391488, 3), 7174442, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(922, 4782976, F9, 4) (dual of [4782976, 4782954, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(922, 4782969, F9, 4) (dual of [4782969, 4782947, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(915, 4782969, F9, 3) (dual of [4782969, 4782954, 4]-code or 4782969-cap in PG(14,9)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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(922, 4782976, F9, 4) (dual of [4782976, 4782954, 5]-code), using
- appending kth column [i] based on linear OOA(922, 2391488, F9, 3, 4) (dual of [(2391488, 3), 7174442, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(922, 2391488, F9, 4, 4) (dual of [(2391488, 4), 9565930, 5]-NRT-code), using
- s-reduction based on digital (18, 22, 2391488)-net over F9, using
- digital (31, 37, 1594328)-net over F9, using
- net defined by OOA [i] based on linear OOA(937, 1594328, F9, 6, 6) (dual of [(1594328, 6), 9565931, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(937, 4782984, F9, 6) (dual of [4782984, 4782947, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(936, 4782969, F9, 6) (dual of [4782969, 4782933, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(922, 4782969, F9, 4) (dual of [4782969, 4782947, 5]-code) (see above)
- 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 Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(937, 4782984, F9, 6) (dual of [4782984, 4782947, 7]-code), using
- net defined by OOA [i] based on linear OOA(937, 1594328, F9, 6, 6) (dual of [(1594328, 6), 9565931, 7]-NRT-code), 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 (18, 22, 1594328)-net over F9, using
- generalized (u, u+v)-construction [i] based on
(150−12, 150, large)-Net over F9 — Digital
Digital (138, 150, large)-net over F9, using
- 95 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
(150−12, 150, large)-Net in Base 9 — Upper bound on s
There is no (138, 150, large)-net in base 9, because
- 10 times m-reduction [i] would yield (138, 140, large)-net in base 9, but