MinT - Best Known (210−53, 210, s)-Nets in Base 4

Best Known (210−53, 210, s)-Nets in Base 4

(210−53, 210, 546)-Net over F₄ — Constructive and digital

Digital (157, 210, 546)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 30, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, 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]

(210−53, 210, 648)-Net in Base 4 — Constructive

(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]

(210−53, 210, 1846)-Net over F₄ — Digital

Digital (157, 210, 1846)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−53, 210, 243091)-Net in Base 4 — Upper bound on s

There is no (157, 210, 243092)-net in base 4, because

1 times m-reduction [i] would yield (157, 209, 243092)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 676946 036245 423796 283825 359508 871489 298435 898415 701668 650905 660859 227250 387342 514861 623178 163864 791467 567962 381133 781969 547856 > 4²⁰⁹ [i]