MinT - Best Known (215−59, 215, s)-Nets in Base 4

Best Known (215−59, 215, s)-Nets in Base 4

(215−59, 215, 531)-Net over F₄ — Constructive and digital

Digital (156, 215, 531)-net over F₄, using

t-expansion [i] based on digital (155, 215, 531)-net over F₄, using
- 7 times m-reduction [i] based on digital (155, 222, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 74, 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]

(215−59, 215, 1298)-Net over F₄ — Digital

Digital (156, 215, 1298)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−59, 215, 107818)-Net in Base 4 — Upper bound on s

There is no (156, 215, 107819)-net in base 4, because

1 times m-reduction [i] would yield (156, 214, 107819)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 693 274559 819004 928814 928583 972658 581497 507089 368244 639701 303701 584205 285829 354135 792519 090105 500123 036015 046540 711446 795197 775760 > 4²¹⁴ [i]