MinT - Best Known (109−29, 109, s)-Nets in Base 7

Best Known (109−29, 109, s)-Nets in Base 7

(109−29, 109, 688)-Net over F₇ — Constructive and digital

Digital (80, 109, 688)-net over F₇, using

t-expansion [i] based on 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]

(109−29, 109, 3684)-Net over F₇ — Digital

Digital (80, 109, 3684)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−29, 109, 3331448)-Net in Base 7 — Upper bound on s

There is no (80, 109, 3331449)-net in base 7, because

1 times m-reduction [i] would yield (80, 108, 3331449)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 18 646170 169563 211891 650061 268313 878235 791165 518311 446540 012089 475422 929521 618548 294757 246017 > 7¹⁰⁸ [i]