MinT - Best Known (205−55, 205, s)-Nets in Base 4

Best Known (205−55, 205, s)-Nets in Base 4

(205−55, 205, 531)-Net over F₄ — Constructive and digital

Digital (150, 205, 531)-net over F₄, using

t-expansion [i] based on digital (149, 205, 531)-net over F₄, using
- 8 times m-reduction [i] based on digital (149, 213, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 71, 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]

(205−55, 205, 576)-Net in Base 4 — Constructive

(150, 205, 576)-net in base 4, using

4¹ times duplication [i] based on (149, 204, 576)-net in base 4, using
- trace code for nets [i] based on (13, 68, 192)-net in base 64, using
  - 2 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]

(205−55, 205, 1376)-Net over F₄ — Digital

Digital (150, 205, 1376)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(205−55, 205, 128857)-Net in Base 4 — Upper bound on s

There is no (150, 205, 128858)-net in base 4, because

1 times m-reduction [i] would yield (150, 204, 128858)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 661 150293 455402 029529 480103 685684 238330 188147 008367 832613 946517 228683 314886 041544 525990 202716 688111 406129 204587 916728 565808 > 4²⁰⁴ [i]