MinT - Best Known (230−93, 230, s)-Nets in Base 4

Best Known (230−93, 230, s)-Nets in Base 4

(230−93, 230, 137)-Net over F₄ — Constructive and digital

Digital (137, 230, 137)-net over F₄, using

5 times m-reduction [i] based on digital (137, 235, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 64, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 171, 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]

(230−93, 230, 319)-Net over F₄ — Digital

Digital (137, 230, 319)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−93, 230, 5923)-Net in Base 4 — Upper bound on s

There is no (137, 230, 5924)-net in base 4, because

1 times m-reduction [i] would yield (137, 229, 5924)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 748467 958825 731638 482375 260276 733541 936920 611489 780006 455304 279742 646718 528367 861157 601274 500879 603361 717060 985584 577835 424705 838584 789696 > 4²²⁹ [i]