MinT - Best Known (112−4, 112, s)-Nets in Base 3

Best Known (112−4, 112, s)-Nets in Base 3

(112−4, 112, large)-Net over F₃ — Constructive and digital

Digital (108, 112, large)-net over F₃, using

3¹² times duplication [i] based on digital (96, 100, large)-net over F₃, using
- t-expansion [i] based on digital (93, 100, large)-net over F₃, using
  - trace code for nets [i] based on digital (43, 50, 5592401)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁵⁰, 5592401, F₉, 9, 7) (dual of [(5592401, 9), 50331559, 8]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(9⁵⁰, 5592402, F₉, 3, 7) (dual of [(5592402, 3), 16777156, 8]-NRT-code), using
        trace code [i] based on linear OOA(81²⁵, 2796201, F₈₁, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(81²⁵, large, F₈₁, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(112−4, 112, large)-Net in Base 3 — Upper bound on s

There is no (108, 112, large)-net in base 3, because

2 times m-reduction [i] would yield (108, 110, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]