MinT - Best Known (58, 83, s)-Nets in Base 7

Best Known (58, 83, s)-Nets in Base 7

(58, 83, 213)-Net over F₇ — Constructive and digital

Digital (58, 83, 213)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 9, 13)-net over F₇, using
  - 4 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (12, 24, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 12, 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]
3. 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 (see above)

(58, 83, 1379)-Net over F₇ — Digital

Digital (58, 83, 1379)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 83, 524856)-Net in Base 7 — Upper bound on s

There is no (58, 83, 524857)-net in base 7, because

1 times m-reduction [i] would yield (58, 82, 524857)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1986 298137 508108 163491 722222 653740 669045 639795 157637 507460 484984 353249 > 7⁸² [i]