MinT - Best Known (159, 159+57, s)-Nets in Base 4

Best Known (159, 159+57, s)-Nets in Base 4

(159, 159+57, 531)-Net over F₄ — Constructive and digital

Digital (159, 216, 531)-net over F₄, using

12 times m-reduction [i] based on digital (159, 228, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 76, 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]

(159, 159+57, 576)-Net in Base 4 — Constructive

(159, 216, 576)-net in base 4, using

t-expansion [i] based on (158, 216, 576)-net in base 4, using
- trace code for nets [i] based on (14, 72, 192)-net in base 64, using
  - 5 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
    - base change [i] based on digital (3, 66, 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]

(159, 159+57, 1547)-Net over F₄ — Digital

Digital (159, 216, 1547)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+57, 158045)-Net in Base 4 — Upper bound on s

There is no (159, 216, 158046)-net in base 4, because

1 times m-reduction [i] would yield (159, 215, 158046)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2772 995424 525932 204653 778372 567868 483077 341127 777601 085640 485172 541036 239507 495092 400535 969177 768066 640214 974365 679390 783949 300593 > 4²¹⁵ [i]