MinT - Best Known (71−31, 71, s)-Nets in Base 8

Best Known (71−31, 71, s)-Nets in Base 8

Digital (40, 71, 208)-net over F₈, using

3 times m-reduction [i] based on digital (40, 74, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 37, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

Digital (40, 71, 258)-net over F₈, using

1 times m-reduction [i] based on digital (40, 72, 258)-net over F₈, using
- trace code for nets [i] based on digital (4, 36, 129)-net over F₆₄, using
  - net from sequence [i] based on digital (4, 128)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 4 and N(F) ≥ 129, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

There is no (40, 71, 15026)-net in base 8, because

1 times m-reduction [i] would yield (40, 70, 15026)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1646 582935 241223 443358 683943 090756 325399 301315 226441 484334 641992 > 8⁷⁰ [i]