MinT - Best Known (78−27, 78, s)-Nets in Base 7

Best Known (78−27, 78, s)-Nets in Base 7

(78−27, 78, 126)-Net over F₇ — Constructive and digital

Digital (51, 78, 126)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 24, 26)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 7, 13)-net over F₇, using
      - 6 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
    2. digital (4, 17, 13)-net over F₇, using
      - Niederreiter sequence (Bratleyâ€“Foxâ€“Niederreiter implementation) with equidistant coordinate [i]
2. digital (27, 54, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 27, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
      - Niederreiter sequence [i]

(78−27, 78, 616)-Net over F₇ — Digital

Digital (51, 78, 616)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−27, 78, 95675)-Net in Base 7 — Upper bound on s

There is no (51, 78, 95676)-net in base 7, because

1 times m-reduction [i] would yield (51, 77, 95676)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 118197 135106 742568 192016 374949 621289 377420 653859 321100 227790 777737 > 7⁷⁷ [i]