MinT - Best Known (171, 171+61, s)-Nets in Base 4

Best Known (171, 171+61, s)-Nets in Base 4

(171, 171+61, 531)-Net over F₄ — Constructive and digital

Digital (171, 232, 531)-net over F₄, using

14 times m-reduction [i] based on digital (171, 246, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 82, 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]

(171, 171+61, 648)-Net in Base 4 — Constructive

(171, 232, 648)-net in base 4, using

4¹ times duplication [i] based on (170, 231, 648)-net in base 4, using
- trace code for nets [i] based on (16, 77, 216)-net in base 64, using
  - base change [i] based on digital (5, 66, 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]

(171, 171+61, 1675)-Net over F₄ — Digital

Digital (171, 232, 1675)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(171, 171+61, 173567)-Net in Base 4 — Upper bound on s

There is no (171, 232, 173568)-net in base 4, because

1 times m-reduction [i] would yield (171, 231, 173568)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 909847 436574 719102 883037 311272 791211 617906 205978 215381 975818 558524 712188 504501 701602 219720 779983 657003 072029 054269 803203 430750 893608 308385 > 4²³¹ [i]