MinT - Best Known (164−42, 164, s)-Nets in Base 4

Best Known (164−42, 164, s)-Nets in Base 4

(164−42, 164, 531)-Net over F₄ — Constructive and digital

Digital (122, 164, 531)-net over F₄, using

t-expansion [i] based on digital (121, 164, 531)-net over F₄, using
- 7 times m-reduction [i] based on digital (121, 171, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 57, 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]

(164−42, 164, 576)-Net in Base 4 — Constructive

(122, 164, 576)-net in base 4, using

t-expansion [i] based on (121, 164, 576)-net in base 4, using
- 1 times m-reduction [i] based on (121, 165, 576)-net in base 4, using
  - trace code for nets [i] based on (11, 55, 192)-net in base 64, using
    - 1 times m-reduction [i] based on (11, 56, 192)-net in base 64, using
      - base change [i] based on digital (3, 48, 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]

(164−42, 164, 1398)-Net over F₄ — Digital

Digital (122, 164, 1398)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−42, 164, 145586)-Net in Base 4 — Upper bound on s

There is no (122, 164, 145587)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 546 816757 195890 925632 634368 480653 437403 299726 963246 898443 122818 137120 060805 735440 361672 763765 751312 > 4¹⁶⁴ [i]