MinT - Best Known (184, 184+63, s)-Nets in Base 4

Best Known (184, 184+63, s)-Nets in Base 4

(184, 184+63, 548)-Net over F₄ — Constructive and digital

Digital (184, 247, 548)-net over F₄, using

4¹ times duplication [i] based on digital (183, 246, 548)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 36, 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 (147, 210, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 70, 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]

(184, 184+63, 648)-Net in Base 4 — Constructive

(184, 247, 648)-net in base 4, using

t-expansion [i] based on (183, 247, 648)-net in base 4, using
- 2 times m-reduction [i] based on (183, 249, 648)-net in base 4, using
  - trace code for nets [i] based on (17, 83, 216)-net in base 64, using
    - 1 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
      - base change [i] based on digital (5, 72, 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]

(184, 184+63, 2028)-Net over F₄ — Digital

Digital (184, 247, 2028)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(184, 184+63, 248030)-Net in Base 4 — Upper bound on s

There is no (184, 247, 248031)-net in base 4, because

1 times m-reduction [i] would yield (184, 246, 248031)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12787 833966 779501 157716 130112 622165 277283 568465 023895 730183 925622 344294 547686 251857 768668 146962 842400 108023 083006 425150 654033 799940 633212 253423 786636 > 4²⁴⁶ [i]