MinT - Best Known (165−53, 165, s)-Nets in Base 4

Best Known (165−53, 165, s)-Nets in Base 4

(165−53, 165, 240)-Net over F₄ — Constructive and digital

Digital (112, 165, 240)-net over F₄, using

t-expansion [i] based on digital (111, 165, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 55, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(165−53, 165, 537)-Net over F₄ — Digital

Digital (112, 165, 537)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−53, 165, 22047)-Net in Base 4 — Upper bound on s

There is no (112, 165, 22048)-net in base 4, because

1 times m-reduction [i] would yield (112, 164, 22048)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 546 970486 916575 906691 909911 784504 874295 874402 398772 257774 503401 785021 314216 974454 226497 502356 521815 > 4¹⁶⁴ [i]