MinT - Best Known (243−64, 243, s)-Nets in Base 4

Best Known (243−64, 243, s)-Nets in Base 4

(243−64, 243, 531)-Net over F₄ — Constructive and digital

Digital (179, 243, 531)-net over F₄, using

15 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]

(243−64, 243, 648)-Net in Base 4 — Constructive

(179, 243, 648)-net in base 4, using

trace code for nets [i] based on (17, 81, 216)-net in base 64, using
- 3 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
  - base change [i] based on digital (5, 72, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(243−64, 243, 1733)-Net over F₄ — Digital

Digital (179, 243, 1733)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−64, 243, 159067)-Net in Base 4 — Upper bound on s

There is no (179, 243, 159068)-net in base 4, because

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]