MinT - Best Known (180, 180+67, s)-Nets in Base 4

Best Known (180, 180+67, s)-Nets in Base 4

(180, 180+67, 531)-Net over F₄ — Constructive and digital

Digital (180, 247, 531)-net over F₄, using

t-expansion [i] based on digital (179, 247, 531)-net over F₄, using
- 11 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]

(180, 180+67, 576)-Net in Base 4 — Constructive

(180, 247, 576)-net in base 4, using

4¹ times duplication [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]

(180, 180+67, 1537)-Net over F₄ — Digital

Digital (180, 247, 1537)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(180, 180+67, 134967)-Net in Base 4 — Upper bound on s

There is no (180, 247, 134968)-net in base 4, because

1 times m-reduction [i] would yield (180, 246, 134968)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12787 894501 822807 143293 079457 187202 491442 710142 852858 687707 068282 329412 052730 946797 387869 423326 248881 783140 530694 710718 104611 867665 734832 442603 265175 > 4²⁴⁶ [i]