MinT - Best Known (59−21, 59, s)-Nets in Base 7

Best Known (59−21, 59, s)-Nets in Base 7

(59−21, 59, 126)-Net over F₇ — Constructive and digital

Digital (38, 59, 126)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 17, 26)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 6, 13)-net over F₇, using
      - 7 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
    2. digital (1, 11, 13)-net over F₇, using
      - 2 times m-reduction [i] based on digital (1, 13, 13)-net over F₇ (see above)
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, 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]

(59−21, 59, 441)-Net over F₇ — Digital

Digital (38, 59, 441)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−21, 59, 60165)-Net in Base 7 — Upper bound on s

There is no (38, 59, 60166)-net in base 7, because

1 times m-reduction [i] would yield (38, 58, 60166)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 10 368261 762514 257905 754570 962628 021448 290261 912589 > 7⁵⁸ [i]