Best Known (126−9, 126, s)-Nets in Base 3
(126−9, 126, 6291450)-Net over F3 — Constructive and digital
Digital (117, 126, 6291450)-net over F3, using
- 33 times duplication [i] based on digital (114, 123, 6291450)-net over F3, using
- trace code for nets [i] based on digital (32, 41, 2097150)-net over F27, using
- net defined by OOA [i] based on linear OOA(2741, 2097150, F27, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2741, 8388601, F27, 9) (dual of [8388601, 8388560, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2741, 8388601, F27, 9) (dual of [8388601, 8388560, 10]-code), using
- net defined by OOA [i] based on linear OOA(2741, 2097150, F27, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- trace code for nets [i] based on digital (32, 41, 2097150)-net over F27, using
(126−9, 126, large)-Net over F3 — Digital
Digital (117, 126, large)-net over F3, using
- t-expansion [i] based on digital (115, 126, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3126, large, F3, 11) (dual of [large, large−126, 12]-code), using
- 20 times code embedding in larger space [i] based on linear OA(3106, large, F3, 11) (dual of [large, large−106, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 20 times code embedding in larger space [i] based on linear OA(3106, large, F3, 11) (dual of [large, large−106, 12]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3126, large, F3, 11) (dual of [large, large−126, 12]-code), using
(126−9, 126, large)-Net in Base 3 — Upper bound on s
There is no (117, 126, large)-net in base 3, because
- 7 times m-reduction [i] would yield (117, 119, large)-net in base 3, but