MinT - Best Known (164−51, 164, s)-Nets in Base 4

Best Known (164−51, 164, s)-Nets in Base 4

(164−51, 164, 312)-Net over F₄ — Constructive and digital

Digital (113, 164, 312)-net over F₄, using

1 times m-reduction [i] based on digital (113, 165, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 55, 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]

(164−51, 164, 607)-Net over F₄ — Digital

Digital (113, 164, 607)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−51, 164, 28551)-Net in Base 4 — Upper bound on s

There is no (113, 164, 28552)-net in base 4, because

1 times m-reduction [i] would yield (113, 163, 28552)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 136 781157 051076 147705 512762 527189 287023 339265 164060 728924 411021 304635 375366 677125 504582 122381 144080 > 4¹⁶³ [i]