MinT - Best Known (81−35, 81, s)-Nets in Base 4

Best Known (81−35, 81, s)-Nets in Base 4

(81−35, 81, 98)-Net over F₄ — Constructive and digital

Digital (46, 81, 98)-net over F₄, using

1 times m-reduction [i] based on digital (46, 82, 98)-net over F₄, using
- trace code for nets [i] based on digital (5, 41, 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]

(81−35, 81, 109)-Net over F₄ — Digital

Digital (46, 81, 109)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−35, 81, 1615)-Net in Base 4 — Upper bound on s

There is no (46, 81, 1616)-net in base 4, because

1 times m-reduction [i] would yield (46, 80, 1616)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 463765 718413 418237 993861 426031 193720 127357 759172 > 4⁸⁰ [i]