MinT - Best Known (165−45, 165, s)-Nets in Base 4

Best Known (165−45, 165, s)-Nets in Base 4

(165−45, 165, 531)-Net over F₄ — Constructive and digital

Digital (120, 165, 531)-net over F₄, using

t-expansion [i] based on digital (119, 165, 531)-net over F₄, using
- 3 times m-reduction [i] based on digital (119, 168, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 56, 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]

(165−45, 165, 1063)-Net over F₄ — Digital

Digital (120, 165, 1063)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−45, 165, 92839)-Net in Base 4 — Upper bound on s

There is no (120, 165, 92840)-net in base 4, because

1 times m-reduction [i] would yield (120, 164, 92840)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 546 901213 526611 531597 467398 320466 790130 290570 559574 569430 184928 871546 566152 684758 474515 964193 443386 > 4¹⁶⁴ [i]