MinT - Best Known (136, 136+99, s)-Nets in Base 4

Best Known (136, 136+99, s)-Nets in Base 4

(136, 136+99, 134)-Net over F₄ — Constructive and digital

Digital (136, 235, 134)-net over F₄, using

1 times m-reduction [i] based on digital (136, 236, 134)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 63, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 173, 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]

(136, 136+99, 285)-Net over F₄ — Digital

Digital (136, 235, 285)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+99, 4738)-Net in Base 4 — Upper bound on s

There is no (136, 235, 4739)-net in base 4, because

1 times m-reduction [i] would yield (136, 234, 4739)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 764 289469 403791 107760 324775 640220 150871 162568 994347 896725 524218 010303 393377 204004 573527 462966 647917 000399 041882 880608 192863 784762 248299 007504 > 4²³⁴ [i]