Best Known (128−12, 128, s)-Nets in Base 3
(128−12, 128, 1398109)-Net over F3 — Constructive and digital
Digital (116, 128, 1398109)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 9)-net over F3, using
- 1 times m-reduction [i] based on digital (2, 9, 9)-net over F3, using
- digital (108, 120, 1398100)-net over F3, using
- net defined by OOA [i] based on linear OOA(3120, 1398100, F3, 12, 12) (dual of [(1398100, 12), 16777080, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3120, 8388600, F3, 12) (dual of [8388600, 8388480, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(3120, 8388600, F3, 12) (dual of [8388600, 8388480, 13]-code), using
- net defined by OOA [i] based on linear OOA(3120, 1398100, F3, 12, 12) (dual of [(1398100, 12), 16777080, 13]-NRT-code), using
- digital (2, 8, 9)-net over F3, using
(128−12, 128, 4194310)-Net over F3 — Digital
Digital (116, 128, 4194310)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3128, 4194310, F3, 2, 12) (dual of [(4194310, 2), 8388492, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(38, 9, F3, 2, 6) (dual of [(9, 2), 10, 7]-NRT-code), using
- extracting embedded OOA [i] based on digital (2, 8, 9)-net over F3, using
- 1 times m-reduction [i] based on digital (2, 9, 9)-net over F3, using
- extracting embedded OOA [i] based on digital (2, 8, 9)-net over F3, using
- linear OOA(3120, 4194301, F3, 2, 12) (dual of [(4194301, 2), 8388482, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3120, 8388602, F3, 12) (dual of [8388602, 8388482, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- OOA 2-folding [i] based on linear OA(3120, 8388602, F3, 12) (dual of [8388602, 8388482, 13]-code), using
- linear OOA(38, 9, F3, 2, 6) (dual of [(9, 2), 10, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
(128−12, 128, large)-Net in Base 3 — Upper bound on s
There is no (116, 128, large)-net in base 3, because
- 10 times m-reduction [i] would yield (116, 118, large)-net in base 3, but