MinT - Best Known (230−58, 230, s)-Nets in Base 4

Best Known (230−58, 230, s)-Nets in Base 4

(230−58, 230, 548)-Net over F₄ — Constructive and digital

Digital (172, 230, 548)-net over F₄, using

4¹ times duplication [i] based on digital (171, 229, 548)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 34, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (137, 195, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 65, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(230−58, 230, 648)-Net in Base 4 — Constructive

(172, 230, 648)-net in base 4, using

t-expansion [i] based on (170, 230, 648)-net in base 4, using
- 1 times m-reduction [i] based on (170, 231, 648)-net in base 4, using
  - trace code for nets [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(230−58, 230, 2006)-Net over F₄ — Digital

Digital (172, 230, 2006)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−58, 230, 231694)-Net in Base 4 — Upper bound on s

There is no (172, 230, 231695)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 977468 126011 815679 188246 848726 543443 683360 807091 554619 378712 442097 028936 709309 980332 385130 376356 596598 576993 349158 056246 228621 909397 939680 > 4²³⁰ [i]