MinT - Best Known (166, 166+61, s)-Nets in Base 4

Best Known (166, 166+61, s)-Nets in Base 4

(166, 166+61, 531)-Net over F₄ — Constructive and digital

Digital (166, 227, 531)-net over F₄, using

t-expansion [i] based on digital (165, 227, 531)-net over F₄, using
- 10 times m-reduction [i] based on digital (165, 237, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 79, 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]

(166, 166+61, 576)-Net in Base 4 — Constructive

(166, 227, 576)-net in base 4, using

1 times m-reduction [i] based on (166, 228, 576)-net in base 4, using
- trace code for nets [i] based on (14, 76, 192)-net in base 64, using
  - 1 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
    - base change [i] based on digital (3, 66, 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]

(166, 166+61, 1495)-Net over F₄ — Digital

Digital (166, 227, 1495)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166, 166+61, 137755)-Net in Base 4 — Upper bound on s

There is no (166, 227, 137756)-net in base 4, because

1 times m-reduction [i] would yield (166, 226, 137756)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11630 813770 560981 559182 921863 418654 793595 857792 523646 694582 748440 047687 976740 500458 349860 592211 911397 556460 821673 062452 805194 074431 539328 > 4²²⁶ [i]