MinT - Best Known (65, 65+57, s)-Nets in Base 4

Best Known (65, 65+57, s)-Nets in Base 4

(65, 65+57, 90)-Net over F₄ — Constructive and digital

Digital (65, 122, 90)-net over F₄, using

trace code for nets [i] based on digital (4, 61, 45)-net over F₁₆, using
- net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
    - a function field by GarcÃa and Quoos [i]

(65, 65+57, 117)-Net over F₄ — Digital

Digital (65, 122, 117)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+57, 1482)-Net in Base 4 — Upper bound on s

There is no (65, 122, 1483)-net in base 4, because

1 times m-reduction [i] would yield (65, 121, 1483)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 111155 408171 268336 993143 911712 563175 609359 812277 961809 691407 842997 106160 > 4¹²¹ [i]