MinT - Best Known (198−53, 198, s)-Nets in Base 4

Best Known (198−53, 198, s)-Nets in Base 4

(198−53, 198, 531)-Net over F₄ — Constructive and digital

Digital (145, 198, 531)-net over F₄, using

9 times m-reduction [i] based on digital (145, 207, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 69, 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]

(198−53, 198, 576)-Net in Base 4 — Constructive

(145, 198, 576)-net in base 4, using

trace code for nets [i] based on (13, 66, 192)-net in base 64, using
- 4 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
  - base change [i] based on digital (3, 60, 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]

(198−53, 198, 1348)-Net over F₄ — Digital

Digital (145, 198, 1348)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(198−53, 198, 128192)-Net in Base 4 — Upper bound on s

There is no (145, 198, 128193)-net in base 4, because

1 times m-reduction [i] would yield (145, 197, 128193)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40352 443783 434333 736390 545080 378127 931574 957905 038394 816949 205472 087864 511202 051835 086748 757590 952832 786123 418613 658400 > 4¹⁹⁷ [i]