MinT - Best Known (256−65, 256, s)-Nets in Base 4

Best Known (256−65, 256, s)-Nets in Base 4

(256−65, 256, 552)-Net over F₄ — Constructive and digital

Digital (191, 256, 552)-net over F₄, using

4¹ times duplication [i] based on digital (190, 255, 552)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 39, 21)-net over F₄, using
    - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (151, 216, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 72, 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]

(256−65, 256, 648)-Net in Base 4 — Constructive

(191, 256, 648)-net in base 4, using

t-expansion [i] based on (190, 256, 648)-net in base 4, using
- 2 times m-reduction [i] based on (190, 258, 648)-net in base 4, using
  - trace code for nets [i] based on (18, 86, 216)-net in base 64, using
    - 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
      - base change [i] based on digital (5, 78, 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]

(256−65, 256, 2137)-Net over F₄ — Digital

Digital (191, 256, 2137)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−65, 256, 267535)-Net in Base 4 — Upper bound on s

There is no (191, 256, 267536)-net in base 4, because

1 times m-reduction [i] would yield (191, 255, 267536)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3351 960444 171512 644607 682763 856234 097956 743596 188877 436898 256927 744577 771323 449602 694682 940709 273330 648777 758515 200613 628073 323665 433153 251603 367855 171000 > 4²⁵⁵ [i]