MinT - Best Known (244−65, 244, s)-Nets in Base 4

Best Known (244−65, 244, s)-Nets in Base 4

(244−65, 244, 531)-Net over F₄ — Constructive and digital

Digital (179, 244, 531)-net over F₄, using

14 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]

(244−65, 244, 576)-Net in Base 4 — Constructive

(179, 244, 576)-net in base 4, using

2 times m-reduction [i] based on (179, 246, 576)-net in base 4, using
- trace code for nets [i] based on (15, 82, 192)-net in base 64, using
  - 2 times m-reduction [i] based on (15, 84, 192)-net in base 64, using
    - base change [i] based on digital (3, 72, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(244−65, 244, 1654)-Net over F₄ — Digital

Digital (179, 244, 1654)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(244−65, 244, 159067)-Net in Base 4 — Upper bound on s

There is no (179, 244, 159068)-net in base 4, because

1 times m-reduction [i] would yield (179, 243, 159068)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 199 822531 112697 803844 995921 554765 160952 095702 212020 089879 790817 560653 927937 214834 871311 372637 424672 237859 152153 460700 013869 653291 436908 312209 995170 > 4²⁴³ [i]