MinT - Best Known (256−109, 256, s)-Nets in Base 4

Best Known (256−109, 256, s)-Nets in Base 4

(256−109, 256, 137)-Net over F₄ — Constructive and digital

Digital (147, 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]

(256−109, 256, 297)-Net over F₄ — Digital

Digital (147, 256, 297)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−109, 256, 4825)-Net in Base 4 — Upper bound on s

There is no (147, 256, 4826)-net in base 4, because

1 times m-reduction [i] would yield (147, 255, 4826)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3384 200835 787459 916116 383928 965881 746487 760617 854362 933034 431462 117839 003914 765198 152868 220083 234889 483360 948569 236752 160061 427507 170643 224489 771587 536200 > 4²⁵⁵ [i]