MinT - Best Known (165, 165+61, s)-Nets in Base 4

Best Known (165, 165+61, s)-Nets in Base 4

(165, 165+61, 531)-Net over F₄ — Constructive and digital

Digital (165, 226, 531)-net over F₄, using

11 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]

(165, 165+61, 576)-Net in Base 4 — Constructive

(165, 226, 576)-net in base 4, using

4¹ times duplication [i] based on (164, 225, 576)-net in base 4, using
- trace code for nets [i] based on (14, 75, 192)-net in base 64, using
  - 2 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]

(165, 165+61, 1460)-Net over F₄ — Digital

Digital (165, 226, 1460)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+61, 131533)-Net in Base 4 — Upper bound on s

There is no (165, 226, 131534)-net in base 4, because

1 times m-reduction [i] would yield (165, 225, 131534)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2907 667795 185245 679968 329885 491602 736988 500067 280508 236596 602453 892942 574775 795610 887523 731276 053392 760362 391756 621670 810548 964527 920384 > 4²²⁵ [i]