MinT - Best Known (89−39, 89, s)-Nets in Base 4

Best Known (89−39, 89, s)-Nets in Base 4

(89−39, 89, 98)-Net over F₄ — Constructive and digital

Digital (50, 89, 98)-net over F₄, using

1 times m-reduction [i] based on digital (50, 90, 98)-net over F₄, using
- trace code for nets [i] based on digital (5, 45, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]

(89−39, 89, 111)-Net over F₄ — Digital

Digital (50, 89, 111)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−39, 89, 1608)-Net in Base 4 — Upper bound on s

There is no (50, 89, 1609)-net in base 4, because

1 times m-reduction [i] would yield (50, 88, 1609)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 96106 764052 610361 002655 208733 996340 526775 241624 099232 > 4⁸⁸ [i]