MinT - Best Known (242−53, 242, s)-Nets in Base 3

Best Known (242−53, 242, s)-Nets in Base 3

(242−53, 242, 688)-Net over F₃ — Constructive and digital

Digital (189, 242, 688)-net over F₃, using

3² times duplication [i] based on digital (187, 240, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 60, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(242−53, 242, 1706)-Net over F₃ — Digital

Digital (189, 242, 1706)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−53, 242, 139550)-Net in Base 3 — Upper bound on s

There is no (189, 242, 139551)-net in base 3, because

1 times m-reduction [i] would yield (189, 241, 139551)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 689036 729473 709889 402999 890299 282386 513426 714052 256999 095266 913627 008036 085728 785450 849723 961851 685128 446543 589901 > 3²⁴¹ [i]