MinT - Best Known (242−55, 242, s)-Nets in Base 3

Best Known (242−55, 242, s)-Nets in Base 3

(242−55, 242, 640)-Net over F₃ — Constructive and digital

Digital (187, 242, 640)-net over F₃, using

3² times duplication [i] based on digital (185, 240, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 60, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(242−55, 242, 1468)-Net over F₃ — Digital

Digital (187, 242, 1468)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−55, 242, 99085)-Net in Base 3 — Upper bound on s

There is no (187, 242, 99086)-net in base 3, because

1 times m-reduction [i] would yield (187, 241, 99086)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 689343 997123 441397 050913 677150 683731 319738 295877 190746 317113 614452 752946 050520 945495 034952 836009 275997 925204 823761 > 3²⁴¹ [i]