MinT - Best Known (109−33, 109, s)-Nets in Base 7

Best Known (109−33, 109, s)-Nets in Base 7

(109−33, 109, 688)-Net over F₇ — Constructive and digital

Digital (76, 109, 688)-net over F₇, using

1 times m-reduction [i] based on digital (76, 110, 688)-net over F₇, using
- trace code for nets [i] based on digital (21, 55, 344)-net over F₄₉, using
  - net from sequence [i] based on digital (21, 343)-sequence over F₄₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344, using
      - the Hermitian function field over F₄₉ [i]

(109−33, 109, 1628)-Net over F₇ — Digital

Digital (76, 109, 1628)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−33, 109, 573840)-Net in Base 7 — Upper bound on s

There is no (76, 109, 573841)-net in base 7, because

1 times m-reduction [i] would yield (76, 108, 573841)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 18 646130 282171 587213 360838 784039 170642 038906 380893 357079 337047 917454 367480 482063 142601 973121 > 7¹⁰⁸ [i]