MinT - Best Known (76−26, 76, s)-Nets in Base 7

Best Known (76−26, 76, s)-Nets in Base 7

(76−26, 76, 126)-Net over F₇ — Constructive and digital

Digital (50, 76, 126)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 24, 26)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 7, 13)-net over F₇, using
      - 6 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
    2. digital (4, 17, 13)-net over F₇, using
      - Niederreiter sequence (Bratleyâ€“Foxâ€“Niederreiter implementation) with equidistant coordinate [i]
2. digital (26, 52, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 26, 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]

(76−26, 76, 641)-Net over F₇ — Digital

Digital (50, 76, 641)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(76−26, 76, 82373)-Net in Base 7 — Upper bound on s

There is no (50, 76, 82374)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 16885 644728 258160 513135 997743 052617 867684 004002 068632 047778 916173 > 7⁷⁶ [i]