MinT - Best Known (165, 165+63, s)-Nets in Base 4

Best Known (165, 165+63, s)-Nets in Base 4

(165, 165+63, 531)-Net over F₄ — Constructive and digital

Digital (165, 228, 531)-net over F₄, using

9 times m-reduction [i] based on digital (165, 237, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 79, 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, 165+63, 1321)-Net over F₄ — Digital

Digital (165, 228, 1321)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+63, 106032)-Net in Base 4 — Upper bound on s

There is no (165, 228, 106033)-net in base 4, because

1 times m-reduction [i] would yield (165, 227, 106033)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46518 486067 987026 647077 371470 492783 129722 997025 991947 022917 802469 013405 921396 163361 143883 165631 743573 344393 400621 407568 463713 566359 844960 > 4²²⁷ [i]