MinT - Best Known (234−94, 234, s)-Nets in Base 4

Best Known (234−94, 234, s)-Nets in Base 4

(234−94, 234, 137)-Net over F₄ — Constructive and digital

Digital (140, 234, 137)-net over F₄, using

10 times m-reduction [i] based on digital (140, 244, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 67, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 177, 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]

(234−94, 234, 331)-Net over F₄ — Digital

Digital (140, 234, 331)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−94, 234, 6049)-Net in Base 4 — Upper bound on s

There is no (140, 234, 6050)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 766 056211 288211 960430 028106 699704 997999 545837 291229 386923 038159 690668 296012 027244 224924 756710 261753 995160 753839 868054 799955 340889 369689 079232 > 4²³⁴ [i]