MinT - Best Known (220−63, 220, s)-Nets in Base 4

Best Known (220−63, 220, s)-Nets in Base 4

(220−63, 220, 531)-Net over F₄ — Constructive and digital

Digital (157, 220, 531)-net over F₄, using

5 times m-reduction [i] based on digital (157, 225, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 75, 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]

(220−63, 220, 1095)-Net over F₄ — Digital

Digital (157, 220, 1095)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−63, 220, 74135)-Net in Base 4 — Upper bound on s

There is no (157, 220, 74136)-net in base 4, because

1 times m-reduction [i] would yield (157, 219, 74136)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 709969 893049 843057 956770 943449 219686 263247 066428 763694 656048 435631 587689 348488 321041 975195 851759 114557 330620 644201 785132 957645 455600 > 4²¹⁹ [i]