MinT - Best Known (219−83, 219, s)-Nets in Base 4

Best Known (219−83, 219, s)-Nets in Base 4

(219−83, 219, 138)-Net over F₄ — Constructive and digital

Digital (136, 219, 138)-net over F₄, using

1 times m-reduction [i] based on digital (136, 220, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 63, 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, 157, 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]

(219−83, 219, 379)-Net over F₄ — Digital

Digital (136, 219, 379)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−83, 219, 8516)-Net in Base 4 — Upper bound on s

There is no (136, 219, 8517)-net in base 4, because

1 times m-reduction [i] would yield (136, 218, 8517)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 177597 143078 329700 970420 361976 583099 204454 558839 800443 556389 014311 448500 752949 100916 473978 535825 202597 604088 504523 438986 008556 522816 > 4²¹⁸ [i]