MinT - Best Known (212−63, 212, s)-Nets in Base 4

Best Known (212−63, 212, s)-Nets in Base 4

(212−63, 212, 531)-Net over F₄ — Constructive and digital

Digital (149, 212, 531)-net over F₄, using

1 times m-reduction [i] based on digital (149, 213, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 71, 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]

(212−63, 212, 907)-Net over F₄ — Digital

Digital (149, 212, 907)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−63, 212, 51831)-Net in Base 4 — Upper bound on s

There is no (149, 212, 51832)-net in base 4, because

1 times m-reduction [i] would yield (149, 211, 51832)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 835562 958660 991387 662633 673602 506495 332920 717465 585370 127737 447385 997804 182490 479561 889503 350972 980484 479710 358645 739527 442932 > 4²¹¹ [i]