MinT - Best Known (139, 139+85, s)-Nets in Base 4

Best Known (139, 139+85, s)-Nets in Base 4

(139, 139+85, 138)-Net over F₄ — Constructive and digital

Digital (139, 224, 138)-net over F₄, using

5 times m-reduction [i] based on digital (139, 229, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 66, 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, 163, 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]

(139, 139+85, 139)-Net in Base 4 — Constructive

(139, 224, 139)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (24, 66, 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, 158, 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]

(139, 139+85, 385)-Net over F₄ — Digital

Digital (139, 224, 385)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+85, 8621)-Net in Base 4 — Upper bound on s

There is no (139, 224, 8622)-net in base 4, because

1 times m-reduction [i] would yield (139, 223, 8622)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 181 836763 741152 885750 862497 777623 820019 050856 104147 760711 361837 555744 921631 108060 755715 811851 284408 150637 476178 990003 134884 338687 118510 > 4²²³ [i]