MinT - Best Known (81−37, 81, s)-Nets in Base 4

Best Known (81−37, 81, s)-Nets in Base 4

(81−37, 81, 76)-Net over F₄ — Constructive and digital

Digital (44, 81, 76)-net over F₄, using

1 times m-reduction [i] based on digital (44, 82, 76)-net over F₄, using
- trace code for nets [i] based on digital (3, 41, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(81−37, 81, 91)-Net over F₄ — Digital

Digital (44, 81, 91)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−37, 81, 1179)-Net in Base 4 — Upper bound on s

There is no (44, 81, 1180)-net in base 4, because

1 times m-reduction [i] would yield (44, 80, 1180)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 481660 660756 525516 228743 771558 422551 647728 562129 > 4⁸⁰ [i]