MinT - Best Known (42, 42+37, s)-Nets in Base 4

Best Known (42, 42+37, s)-Nets in Base 4

(42, 42+37, 66)-Net over F₄ — Constructive and digital

Digital (42, 79, 66)-net over F₄, using

1 times m-reduction [i] based on digital (42, 80, 66)-net over F₄, using
- trace code for nets [i] based on digital (2, 40, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]

(42, 42+37, 82)-Net over F₄ — Digital

Digital (42, 79, 82)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+37, 1008)-Net in Base 4 — Upper bound on s

There is no (42, 79, 1009)-net in base 4, because

1 times m-reduction [i] would yield (42, 78, 1009)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 91795 039806 403321 783742 270966 695976 949675 502816 > 4⁷⁸ [i]