MinT - Best Known (256−110, 256, s)-Nets in Base 4

Best Known (256−110, 256, s)-Nets in Base 4

(256−110, 256, 137)-Net over F₄ — Constructive and digital

Digital (146, 256, 137)-net over F₄, using

t-expansion [i] based on digital (145, 256, 137)-net over F₄, using
- 3 times m-reduction [i] based on digital (145, 259, 137)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 72, 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, 187, 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]

(256−110, 256, 289)-Net over F₄ — Digital

Digital (146, 256, 289)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−110, 256, 4466)-Net in Base 4 — Upper bound on s

There is no (146, 256, 4467)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13462 811137 179641 615074 927701 409261 042249 014564 053373 783389 912657 542213 829771 930799 653008 733859 853878 656452 515135 835653 697533 013908 504127 741482 313852 858240 > 4²⁵⁶ [i]