MinT - Best Known (75, 75+31, s)-Nets in Base 4

Best Known (75, 75+31, s)-Nets in Base 4

(75, 75+31, 312)-Net over F₄ — Constructive and digital

Digital (75, 106, 312)-net over F₄, using

2 times m-reduction [i] based on digital (75, 108, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 36, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(75, 75+31, 553)-Net over F₄ — Digital

Digital (75, 106, 553)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(75, 75+31, 35068)-Net in Base 4 — Upper bound on s

There is no (75, 106, 35069)-net in base 4, because

1 times m-reduction [i] would yield (75, 105, 35069)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1645 880814 384941 748564 954637 618328 470470 127966 016555 523702 446992 > 4¹⁰⁵ [i]