MinT - Best Known (55, 55+28, s)-Nets in Base 7

Best Known (55, 55+28, s)-Nets in Base 7

(55, 55+28, 150)-Net over F₇ — Constructive and digital

Digital (55, 83, 150)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (13, 27, 50)-net over F₇, using
  - base reduction for projective spaces (embedding PG(13,49) in PG(26,7)) for nets [i] based on digital (0, 14, 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 (28, 56, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 28, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(55, 55+28, 735)-Net over F₇ — Digital

Digital (55, 83, 735)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 55+28, 103155)-Net in Base 7 — Upper bound on s

There is no (55, 83, 103156)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 13904 413385 166098 782886 140456 168080 781809 973126 114003 115883 527138 229913 > 7⁸³ [i]