MinT - Best Known (212−61, 212, s)-Nets in Base 4

Best Known (212−61, 212, s)-Nets in Base 4

(212−61, 212, 531)-Net over F₄ — Constructive and digital

Digital (151, 212, 531)-net over F₄, using

4 times m-reduction [i] based on digital (151, 216, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 72, 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]

(212−61, 212, 1040)-Net over F₄ — Digital

Digital (151, 212, 1040)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−61, 212, 68865)-Net in Base 4 — Upper bound on s

There is no (151, 212, 68866)-net in base 4, because

1 times m-reduction [i] would yield (151, 211, 68866)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 833206 908330 805160 653008 486744 623931 145166 770049 364079 798317 572047 841465 156581 163794 198222 551442 710846 600678 385191 579061 469568 > 4²¹¹ [i]