MinT - Best Known (211−53, 211, s)-Nets in Base 4

Best Known (211−53, 211, s)-Nets in Base 4

(211−53, 211, 548)-Net over F₄ — Constructive and digital

Digital (158, 211, 548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 31, 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 (127, 180, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 60, 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]

(211−53, 211, 648)-Net in Base 4 — Constructive

(158, 211, 648)-net in base 4, using

4¹ times duplication [i] based on (157, 210, 648)-net in base 4, using
- t-expansion [i] based on (155, 210, 648)-net in base 4, using
  - trace code for nets [i] based on (15, 70, 216)-net in base 64, using
    - base change [i] based on digital (5, 60, 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]

(211−53, 211, 1895)-Net over F₄ — Digital

Digital (158, 211, 1895)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−53, 211, 256405)-Net in Base 4 — Upper bound on s

There is no (158, 211, 256406)-net in base 4, because

1 times m-reduction [i] would yield (158, 210, 256406)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 707686 899759 238953 395340 080181 982527 522026 026507 165339 531019 672652 371118 988373 419233 033774 767998 507952 099343 472727 033528 047328 > 4²¹⁰ [i]