MinT - Best Known (127, 127+53, s)-Nets in Base 4

Best Known (127, 127+53, s)-Nets in Base 4

(127, 127+53, 531)-Net over F₄ — Constructive and digital

Digital (127, 180, 531)-net over F₄, using

trace code for nets [i] based on digital (7, 60, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(127, 127+53, 822)-Net over F₄ — Digital

Digital (127, 180, 822)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+53, 49083)-Net in Base 4 — Upper bound on s

There is no (127, 180, 49084)-net in base 4, because

1 times m-reduction [i] would yield (127, 179, 49084)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 587272 564729 970691 943903 813200 158508 161489 332728 037397 631596 277915 631140 633438 632137 780418 891531 737955 605729 > 4¹⁷⁹ [i]