MinT - Best Known (222−86, 222, s)-Nets in Base 4

Best Known (222−86, 222, s)-Nets in Base 4

(222−86, 222, 137)-Net over F₄ — Constructive and digital

Digital (136, 222, 137)-net over F₄, using

10 times m-reduction [i] based on digital (136, 232, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 63, 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, 169, 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]

(222−86, 222, 356)-Net over F₄ — Digital

Digital (136, 222, 356)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−86, 222, 7186)-Net in Base 4 — Upper bound on s

There is no (136, 222, 7187)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 501971 251160 602768 556560 676264 967792 426639 817835 480790 799113 027039 588716 371295 213075 420478 109402 784353 591413 970754 561720 039638 485760 > 4²²² [i]