MinT - Best Known (256−96, 256, s)-Nets in Base 4

Best Known (256−96, 256, s)-Nets in Base 4

(256−96, 256, 160)-Net over F₄ — Constructive and digital

Digital (160, 256, 160)-net over F₄, using

t-expansion [i] based on digital (157, 256, 160)-net over F₄, using
- 3 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 56)-net over F₄, using
      - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
    2. digital (73, 175, 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]

(256−96, 256, 452)-Net over F₄ — Digital

Digital (160, 256, 452)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−96, 256, 10114)-Net in Base 4 — Upper bound on s

There is no (160, 256, 10115)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13409 143795 772011 966784 321254 558573 443370 565975 247213 789424 401584 420008 436071 873225 338218 755349 008923 219456 626848 448863 883862 173545 633556 895983 517046 914616 > 4²⁵⁶ [i]