MinT - Best Known (54−18, 54, s)-Nets in Base 7

Best Known (54−18, 54, s)-Nets in Base 7

(54−18, 54, 200)-Net over F₇ — Constructive and digital

Digital (36, 54, 200)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 18, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 9, 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 (18, 36, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 18, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(54−18, 54, 587)-Net over F₇ — Digital

Digital (36, 54, 587)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−18, 54, 81312)-Net in Base 7 — Upper bound on s

There is no (36, 54, 81313)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 4318 196945 927358 635417 618688 394020 628639 006471 > 7⁵⁴ [i]