MinT - Best Known (191−3, 191, s)-Nets in Base 3

Best Known (191−3, 191, s)-Nets in Base 3

(191−3, 191, large)-Net over F₃ — Constructive and digital

Digital (188, 191, large)-net over F₃, using

8 times m-reduction [i] based on digital (188, 199, 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 (142, 153, 5033160)-net over F₃, using
    - trace code for nets [i] based on digital (40, 51, 1677720)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27⁵¹, 1677720, F₂₇, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
        OOA 5-folding and stacking with additional row [i] based on linear OA(27⁵¹, 8388601, F₂₇, 11) (dual of [8388601, 8388550, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁵¹, large, F₂₇, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(191−3, 191, large)-Net in Base 3 — Upper bound on s

There is no (188, 191, large)-net in base 3, because

1 times m-reduction [i] would yield (188, 190, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]