MinT - Best Known (51, 51+25, s)-Nets in Base 7

Best Known (51, 51+25, s)-Nets in Base 7

(51, 51+25, 202)-Net over F₇ — Constructive and digital

Digital (51, 76, 202)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (27, 52, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 26, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(51, 51+25, 787)-Net over F₇ — Digital

Digital (51, 76, 787)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+25, 168676)-Net in Base 7 — Upper bound on s

There is no (51, 76, 168677)-net in base 7, because

1 times m-reduction [i] would yield (51, 75, 168677)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 2412 015949 431202 311043 925422 273744 443332 823307 016035 952585 407777 > 7⁷⁵ [i]