MinT - Best Known (212−57, 212, s)-Nets in Base 4

Best Known (212−57, 212, s)-Nets in Base 4

(212−57, 212, 531)-Net over F₄ — Constructive and digital

Digital (155, 212, 531)-net over F₄, using

10 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]

(212−57, 212, 576)-Net in Base 4 — Constructive

(155, 212, 576)-net in base 4, using

4² times duplication [i] based on (153, 210, 576)-net in base 4, using
- trace code for nets [i] based on (13, 70, 192)-net in base 64, using
  - base change [i] based on digital (3, 60, 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]

(212−57, 212, 1404)-Net over F₄ — Digital

Digital (155, 212, 1404)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−57, 212, 129646)-Net in Base 4 — Upper bound on s

There is no (155, 212, 129647)-net in base 4, because

1 times m-reduction [i] would yield (155, 211, 129647)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 832902 607857 554068 690275 188644 437068 634303 893283 224757 858843 366854 453230 039686 050113 968135 177060 732984 721424 172607 107702 401322 > 4²¹¹ [i]