MinT - Best Known (249−69, 249, s)-Nets in Base 4

Best Known (249−69, 249, s)-Nets in Base 4

(249−69, 249, 531)-Net over F₄ — Constructive and digital

Digital (180, 249, 531)-net over F₄, using

t-expansion [i] based on digital (179, 249, 531)-net over F₄, using
- 9 times m-reduction [i] based on digital (179, 258, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 86, 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]

(249−69, 249, 1405)-Net over F₄ — Digital

Digital (180, 249, 1405)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−69, 249, 111107)-Net in Base 4 — Upper bound on s

There is no (180, 249, 111108)-net in base 4, because

1 times m-reduction [i] would yield (180, 248, 111108)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204643 263155 935087 411472 721229 566001 479962 677728 078100 885914 437627 891085 558585 577890 589291 388536 472537 246201 019086 842887 078683 861222 943653 168131 864912 > 4²⁴⁸ [i]