MinT - Best Known (139, 139+87, s)-Nets in Base 4

Best Known (139, 139+87, s)-Nets in Base 4

(139, 139+87, 138)-Net over F₄ — Constructive and digital

Digital (139, 226, 138)-net over F₄, using

3 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+87, 370)-Net over F₄ — Digital

Digital (139, 226, 370)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+87, 7920)-Net in Base 4 — Upper bound on s

There is no (139, 226, 7921)-net in base 4, because

1 times m-reduction [i] would yield (139, 225, 7921)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2921 354140 445505 137667 032828 231902 969376 894005 483588 329744 090201 448870 912361 012652 180548 751608 886218 829949 176502 949706 111775 866137 284048 > 4²²⁵ [i]