MinT - Best Known (186−10, 186, s)-Nets in Base 3

Best Known (186−10, 186, s)-Nets in Base 3

(186−10, 186, large)-Net over F₃ — Constructive and digital

Digital (176, 186, large)-net over F₃, using

3² times duplication [i] based on digital (174, 184, large)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (41, 46, 4194301)-net over F₃, using
    - net defined by OOA [i] based on linear OOA(3⁴⁶, 4194301, F₃, 5, 5) (dual of [(4194301, 5), 20971459, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(3⁴⁶, large, F₃, 5) (dual of [large, large−46, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
  2. digital (128, 138, 5033160)-net over F₃, using
    - trace code for nets [i] based on digital (36, 46, 1677720)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27⁴⁶, 1677720, F₂₇, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
        OA 5-folding and stacking [i] based on linear OA(27⁴⁶, 8388600, F₂₇, 10) (dual of [8388600, 8388554, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁴⁶, large, F₂₇, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(186−10, 186, large)-Net in Base 3 — Upper bound on s

There is no (176, 186, large)-net in base 3, because

8 times m-reduction [i] would yield (176, 178, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]