MinT - Best Known (170−43, 170, s)-Nets in Base 4

Best Known (170−43, 170, s)-Nets in Base 4

(170−43, 170, 531)-Net over F₄ — Constructive and digital

Digital (127, 170, 531)-net over F₄, using

10 times m-reduction [i] based on 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]

(170−43, 170, 648)-Net in Base 4 — Constructive

(127, 170, 648)-net in base 4, using

4² 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]

(170−43, 170, 1526)-Net over F₄ — Digital

Digital (127, 170, 1526)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−43, 170, 202527)-Net in Base 4 — Upper bound on s

There is no (127, 170, 202528)-net in base 4, because

1 times m-reduction [i] would yield (127, 169, 202528)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 559956 736398 026120 168704 970025 687305 364873 505712 538791 358739 820304 997696 052748 608190 382237 466641 786315 > 4¹⁶⁹ [i]