MinT - Best Known (61, 61+31, s)-Nets in Base 7

Best Known (61, 61+31, s)-Nets in Base 7

(61, 61+31, 200)-Net over F₇ — Constructive and digital

Digital (61, 92, 200)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (15, 30, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 15, 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 (31, 62, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 31, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(61, 61+31, 799)-Net over F₇ — Digital

Digital (61, 92, 799)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(61, 61+31, 143388)-Net in Base 7 — Upper bound on s

There is no (61, 92, 143389)-net in base 7, because

1 times m-reduction [i] would yield (61, 91, 143389)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 80156 410805 544794 437779 205553 544734 832596 667664 705340 733948 762126 805939 181143 > 7⁹¹ [i]