MinT - Best Known (234−61, 234, s)-Nets in Base 4

Best Known (234−61, 234, s)-Nets in Base 4

(234−61, 234, 536)-Net over F₄ — Constructive and digital

Digital (173, 234, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 30, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (143, 204, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 68, 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]

(234−61, 234, 648)-Net in Base 4 — Constructive

(173, 234, 648)-net in base 4, using

4³ times duplication [i] based on (170, 231, 648)-net in base 4, using
- trace code for nets [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]

(234−61, 234, 1753)-Net over F₄ — Digital

Digital (173, 234, 1753)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−61, 234, 190375)-Net in Base 4 — Upper bound on s

There is no (173, 234, 190376)-net in base 4, because

1 times m-reduction [i] would yield (173, 233, 190376)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 190 554855 529603 859868 685374 852949 686612 247968 090543 142995 581820 850865 575597 520439 566226 482323 472472 779270 083658 901965 028958 856842 847854 594156 > 4²³³ [i]