MinT - Best Known (169−43, 169, s)-Nets in Base 4

Best Known (169−43, 169, s)-Nets in Base 4

(169−43, 169, 531)-Net over F₄ — Constructive and digital

Digital (126, 169, 531)-net over F₄, using

t-expansion [i] based on digital (125, 169, 531)-net over F₄, using
- 8 times m-reduction [i] based on digital (125, 177, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 59, 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]

(169−43, 169, 648)-Net in Base 4 — Constructive

(126, 169, 648)-net in base 4, using

4¹ times duplication [i] based on (125, 168, 648)-net in base 4, using
- trace code for nets [i] based on (13, 56, 216)-net in base 64, using
  - base change [i] based on digital (5, 48, 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]

(169−43, 169, 1477)-Net over F₄ — Digital

Digital (126, 169, 1477)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−43, 169, 189588)-Net in Base 4 — Upper bound on s

There is no (126, 169, 189589)-net in base 4, because

1 times m-reduction [i] would yield (126, 168, 189589)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 139989 676867 309103 024129 188751 594098 126838 403013 301710 347896 399472 801748 956695 882567 053410 505000 391668 > 4¹⁶⁸ [i]