Best Known (125, 125+12, s)-Nets in Base 3
(125, 125+12, 1398153)-Net over F3 — Constructive and digital
Digital (125, 137, 1398153)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (11, 17, 53)-net over F3, using
- net defined by OOA [i] based on linear OOA(317, 53, F3, 6, 6) (dual of [(53, 6), 301, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(317, 53, F3, 5, 6) (dual of [(53, 5), 248, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(35, 20, F3, 5, 3) (dual of [(20, 5), 95, 4]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(35, 20, F3, 3, 3) (dual of [(20, 3), 55, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(35, 20, F3, 2, 3) (dual of [(20, 2), 35, 4]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(35, 20, F3, 3, 3) (dual of [(20, 3), 55, 4]-NRT-code), using
- linear OOA(312, 33, F3, 5, 6) (dual of [(33, 5), 153, 7]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 12, 33)-net over F3, using
- linear OOA(35, 20, F3, 5, 3) (dual of [(20, 5), 95, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(317, 53, F3, 5, 6) (dual of [(53, 5), 248, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(317, 53, F3, 6, 6) (dual of [(53, 6), 301, 7]-NRT-code), 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 (11, 17, 53)-net over F3, using
(125, 125+12, 6979026)-Net over F3 — Digital
Digital (125, 137, 6979026)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3137, 6979026, F3, 12) (dual of [6979026, 6978889, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3137, large, F3, 12) (dual of [large, large−137, 13]-code), using
- 17 times code embedding in larger space [i] 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]
- 17 times code embedding in larger space [i] based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3137, large, F3, 12) (dual of [large, large−137, 13]-code), using
(125, 125+12, large)-Net in Base 3 — Upper bound on s
There is no (125, 137, large)-net in base 3, because
- 10 times m-reduction [i] would yield (125, 127, large)-net in base 3, but