MinT - Best Known (86−28, 86, s)-Nets in Base 7

Best Known (86−28, 86, s)-Nets in Base 7

(86−28, 86, 202)-Net over F₇ — Constructive and digital

Digital (58, 86, 202)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (14, 28, 100)-net over F₇, using
  - trace code 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 (30, 58, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 29, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(86−28, 86, 909)-Net over F₇ — Digital

Digital (58, 86, 909)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−28, 86, 156530)-Net in Base 7 — Upper bound on s

There is no (58, 86, 156531)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 4 769329 044515 195696 627346 169794 064449 616996 258657 375339 502155 303636 149013 > 7⁸⁶ [i]