MinT - Best Known (180−35, 180, s)-Nets in Base 4

Best Known (180−35, 180, s)-Nets in Base 4

(180−35, 180, 1104)-Net over F₄ — Constructive and digital

Digital (145, 180, 1104)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (23, 40, 76)-net over F₄, using
  - trace code for nets [i] based on digital (3, 20, 38)-net over F₁₆, using
    - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (105, 140, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 35, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(180−35, 180, 6963)-Net over F₄ — Digital

Digital (145, 180, 6963)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−35, 180, 5225891)-Net in Base 4 — Upper bound on s

There is no (145, 180, 5225892)-net in base 4, because

1 times m-reduction [i] would yield (145, 179, 5225892)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 587136 982539 073717 813890 696473 080000 858477 941809 153086 048210 206740 436805 773315 073217 144647 104660 619615 211590 > 4¹⁷⁹ [i]