MinT - Best Known (42, 42+35, s)-Nets in Base 3

Best Known (42, 42+35, s)-Nets in Base 3

(42, 42+35, 56)-Net over F₃ — Constructive and digital

Digital (42, 77, 56)-net over F₃, using

1 times m-reduction [i] based on digital (42, 78, 56)-net over F₃, using
- trace code for nets [i] based on digital (3, 39, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]

(42, 42+35, 61)-Net over F₃ — Digital

Digital (42, 77, 61)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+35, 471)-Net in Base 3 — Upper bound on s

There is no (42, 77, 472)-net in base 3, because

1 times m-reduction [i] would yield (42, 76, 472)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 877899 269371 456418 388630 970013 479601 > 3⁷⁶ [i]