MinT - Best Known (222−84, 222, s)-Nets in Base 4

Best Known (222−84, 222, s)-Nets in Base 4

(222−84, 222, 138)-Net over F₄ — Constructive and digital

Digital (138, 222, 138)-net over F₄, using

4 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]

(222−84, 222, 386)-Net over F₄ — Digital

Digital (138, 222, 386)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−84, 222, 8340)-Net in Base 4 — Upper bound on s

There is no (138, 222, 8341)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 472344 465565 677284 071949 631241 379921 991196 773968 595878 047437 994127 178925 648372 718973 071373 337082 957800 379917 134248 179741 804250 610736 > 4²²² [i]