MinT - Best Known (68−25, 68, s)-Nets in Base 7

Best Known (68−25, 68, s)-Nets in Base 7

(68−25, 68, 120)-Net over F₇ — Constructive and digital

Digital (43, 68, 120)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 18, 20)-net over F₇, using
  - 2 times m-reduction [i] based on digital (6, 20, 20)-net over F₇, using
    - a shift-net [i]
2. digital (25, 50, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 25, 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]

(68−25, 68, 417)-Net over F₇ — Digital

Digital (43, 68, 417)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−25, 68, 46089)-Net in Base 7 — Upper bound on s

There is no (43, 68, 46090)-net in base 7, because

1 times m-reduction [i] would yield (43, 67, 46090)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 418 414253 242006 093244 701352 865240 371223 508649 127571 277593 > 7⁶⁷ [i]