MinT - Best Known (41, 62, s)-Nets in Base 7

Best Known (41, 62, s)-Nets in Base 7

(41, 62, 200)-Net over F₇ — Constructive and digital

Digital (41, 62, 200)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 20, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 10, 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]
2. digital (21, 42, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 21, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(41, 62, 587)-Net over F₇ — Digital

Digital (41, 62, 587)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(41, 62, 107869)-Net in Base 7 — Upper bound on s

There is no (41, 62, 107870)-net in base 7, because

1 times m-reduction [i] would yield (41, 61, 107870)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 3556 350215 435083 038650 671859 603830 890846 173460 638749 > 7⁶¹ [i]