MinT - Best Known (232−91, 232, s)-Nets in Base 4

Best Known (232−91, 232, s)-Nets in Base 4

(232−91, 232, 138)-Net over F₄ — Constructive and digital

Digital (141, 232, 138)-net over F₄, using

3 times m-reduction [i] based on digital (141, 235, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 68, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 167, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(232−91, 232, 355)-Net over F₄ — Digital

Digital (141, 232, 355)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−91, 232, 7201)-Net in Base 4 — Upper bound on s

There is no (141, 232, 7202)-net in base 4, because

1 times m-reduction [i] would yield (141, 231, 7202)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 967006 869290 302906 684753 976637 909126 743646 226397 716535 503054 182767 426458 993239 359895 312716 816735 143173 074461 800821 575490 941039 878302 160128 > 4²³¹ [i]