MinT - Best Known (227−86, 227, s)-Nets in Base 4

Best Known (227−86, 227, s)-Nets in Base 4

(227−86, 227, 138)-Net over F₄ — Constructive and digital

Digital (141, 227, 138)-net over F₄, using

8 times m-reduction [i] based on digital (141, 235, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 68, 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, 167, 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]

(227−86, 227, 139)-Net in Base 4 — Constructive

(141, 227, 139)-net in base 4, using

2 times m-reduction [i] based on (141, 229, 139)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (24, 68, 35)-net in base 4, using
    - net from sequence [i] based on (24, 34)-sequence in base 4, using
      - base expansion [i] based on digital (48, 34)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [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]

(227−86, 227, 392)-Net over F₄ — Digital

Digital (141, 227, 392)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−86, 227, 8449)-Net in Base 4 — Upper bound on s

There is no (141, 227, 8450)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46527 157548 727272 888652 737290 871424 634230 512643 043494 156922 592538 953735 046961 220316 502047 945447 357822 818457 687533 918859 787719 791843 199360 > 4²²⁷ [i]