MinT - Best Known (94−14, 94, s)-Nets in Base 7

Best Known (94−14, 94, s)-Nets in Base 7

(94−14, 94, 117665)-Net over F₇ — Constructive and digital

Digital (80, 94, 117665)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 10, 16)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 8)-net over F₇, using
      - net from sequence [i] based on digital (0, 7)-sequence over F₇, using
        Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using
        the rational function field F₇(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 7, 8)-net over F₇, using
      - net from sequence [i] based on digital (0, 7)-sequence over F₇ (see above)
2. digital (70, 84, 117649)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁸⁴, 117649, F₇, 14, 14) (dual of [(117649, 14), 1647002, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(7⁸⁴, 823543, F₇, 14) (dual of [823543, 823459, 15]-code), using
      - 1 times truncation [i] based on linear OA(7⁸⁵, 823544, F₇, 15) (dual of [823544, 823459, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 823544Â | 7¹⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(94−14, 94, 1218894)-Net over F₇ — Digital

Digital (80, 94, 1218894)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(94−14, 94, large)-Net in Base 7 — Upper bound on s

There is no (80, 94, large)-net in base 7, because

12 times m-reduction [i] would yield (80, 82, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]