MinT - Best Known (165, 165+57, s)-Nets in Base 4

Best Known (165, 165+57, s)-Nets in Base 4

(165, 165+57, 541)-Net over F₄ — Constructive and digital

Digital (165, 222, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 30, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (135, 192, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 64, 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+57, 648)-Net in Base 4 — Constructive

(165, 222, 648)-net in base 4, using

t-expansion [i] based on (164, 222, 648)-net in base 4, using
- trace code for nets [i] based on (16, 74, 216)-net in base 64, using
  - 3 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(165, 165+57, 1791)-Net over F₄ — Digital

Digital (165, 222, 1791)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+57, 212721)-Net in Base 4 — Upper bound on s

There is no (165, 222, 212722)-net in base 4, because

1 times m-reduction [i] would yield (165, 221, 212722)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 358130 758096 917832 604061 049305 961772 675163 208916 039111 571998 237102 615930 365229 601276 528938 895562 173498 274468 348511 073612 572383 763864 > 4²²¹ [i]