MinT - Best Known (177−45, 177, s)-Nets in Base 4

Best Known (177−45, 177, s)-Nets in Base 4

(177−45, 177, 531)-Net over F₄ — Constructive and digital

Digital (132, 177, 531)-net over F₄, using

t-expansion [i] based on digital (131, 177, 531)-net over F₄, using
- 9 times m-reduction [i] based on digital (131, 186, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 62, 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]

(177−45, 177, 648)-Net in Base 4 — Constructive

(132, 177, 648)-net in base 4, using

trace code for nets [i] based on (14, 59, 216)-net in base 64, using
- 4 times m-reduction [i] based on (14, 63, 216)-net in base 64, using
  - base change [i] based on digital (5, 54, 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]

(177−45, 177, 1541)-Net over F₄ — Digital

Digital (132, 177, 1541)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(177−45, 177, 197775)-Net in Base 4 — Upper bound on s

There is no (132, 177, 197776)-net in base 4, because

1 times m-reduction [i] would yield (132, 176, 197776)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9174 479265 395053 136554 493709 585740 891922 149106 387927 672244 940739 535447 634608 424881 186124 591126 807132 009896 > 4¹⁷⁶ [i]