Best Known (137−10, 137, s)-Nets in Base 3
(137−10, 137, 3355451)-Net over F3 — Constructive and digital
Digital (127, 137, 3355451)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 11)-net over F3, using
- digital (120, 130, 3355440)-net over F3, using
- trace code for nets [i] based on digital (55, 65, 1677720)-net over F9, using
- net defined by OOA [i] based on linear OOA(965, 1677720, F9, 10, 10) (dual of [(1677720, 10), 16777135, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(965, 8388600, F9, 10) (dual of [8388600, 8388535, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(965, large, F9, 10) (dual of [large, large−65, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(965, large, F9, 10) (dual of [large, large−65, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(965, 8388600, F9, 10) (dual of [8388600, 8388535, 11]-code), using
- net defined by OOA [i] based on linear OOA(965, 1677720, F9, 10, 10) (dual of [(1677720, 10), 16777135, 11]-NRT-code), using
- trace code for nets [i] based on digital (55, 65, 1677720)-net over F9, using
(137−10, 137, large)-Net over F3 — Digital
Digital (127, 137, large)-net over F3, using
- 2 times m-reduction [i] based on digital (127, 139, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3139, large, F3, 12) (dual of [large, large−139, 13]-code), using
- 19 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]
- 19 times code embedding in larger space [i] based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3139, large, F3, 12) (dual of [large, large−139, 13]-code), using
(137−10, 137, large)-Net in Base 3 — Upper bound on s
There is no (127, 137, large)-net in base 3, because
- 8 times m-reduction [i] would yield (127, 129, large)-net in base 3, but