MinT - Best Known (70−31, 70, s)-Nets in Base 8

Best Known (70−31, 70, s)-Nets in Base 8

(70−31, 70, 208)-Net over F₈ — Constructive and digital

Digital (39, 70, 208)-net over F₈, using

2 times m-reduction [i] based on digital (39, 72, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 36, 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]

(70−31, 70, 258)-Net over F₈ — Digital

Digital (39, 70, 258)-net over F₈, using

trace code for nets [i] based on digital (4, 35, 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]

(70−31, 70, 13079)-Net in Base 8 — Upper bound on s

There is no (39, 70, 13080)-net in base 8, because

1 times m-reduction [i] would yield (39, 69, 13080)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 205 891348 516121 590385 442405 751355 762991 396544 545315 479926 387542 > 8⁶⁹ [i]