MinT - Best Known (206−55, 206, s)-Nets in Base 4

Best Known (206−55, 206, s)-Nets in Base 4

(206−55, 206, 531)-Net over F₄ — Constructive and digital

Digital (151, 206, 531)-net over F₄, using

10 times m-reduction [i] based on digital (151, 216, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 72, 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]

(206−55, 206, 576)-Net in Base 4 — Constructive

(151, 206, 576)-net in base 4, using

1 times m-reduction [i] based on (151, 207, 576)-net in base 4, using
- trace code for nets [i] based on (13, 69, 192)-net in base 64, using
  - 1 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]

(206−55, 206, 1411)-Net over F₄ — Digital

Digital (151, 206, 1411)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(206−55, 206, 135647)-Net in Base 4 — Upper bound on s

There is no (151, 206, 135648)-net in base 4, because

1 times m-reduction [i] would yield (151, 205, 135648)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2644 564981 128390 995153 672705 503554 038606 111544 686690 361182 667543 672367 630930 395279 627557 426043 523209 023563 290037 555030 218187 > 4²⁰⁵ [i]