MinT - Best Known (93−4, 93, s)-Nets in Base 3

Best Known (93−4, 93, s)-Nets in Base 3

(93−4, 93, large)-Net over F₃ — Constructive and digital

Digital (89, 93, large)-net over F₃, using

3 times m-reduction [i] based on digital (89, 96, large)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 3, 4)-net over F₃, using
    - net from sequence [i] based on digital (0, 3)-sequence over F₃, using
      - Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 0 and N(F) ≥ 4, using
        the rational function field F₃(x) [i]
      - Niederreiter sequence [i]
  2. digital (86, 93, 8388600)-net over F₃, using
    - trace code for nets [i] based on digital (24, 31, 2796200)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27³¹, 2796200, F₂₇, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(27³¹, 8388601, F₂₇, 7) (dual of [8388601, 8388570, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(27³¹, large, F₂₇, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(93−4, 93, large)-Net in Base 3 — Upper bound on s

There is no (89, 93, large)-net in base 3, because

2 times m-reduction [i] would yield (89, 91, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]