MinT - Best Known (224−86, 224, s)-Nets in Base 4

Best Known (224−86, 224, s)-Nets in Base 4

(224−86, 224, 138)-Net over F₄ — Constructive and digital

Digital (138, 224, 138)-net over F₄, using

2 times m-reduction [i] based on digital (138, 226, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 65, 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, 161, 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]

(224−86, 224, 370)-Net over F₄ — Digital

Digital (138, 224, 370)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−86, 224, 7667)-Net in Base 4 — Upper bound on s

There is no (138, 224, 7668)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 727 948782 032188 130932 295730 543002 953593 934013 629146 896336 692900 315799 841922 013426 902529 559121 903958 785173 780077 536807 366625 566372 375690 > 4²²⁴ [i]