MinT - Best Known (90−41, 90, s)-Nets in Base 8

Best Known (90−41, 90, s)-Nets in Base 8

(90−41, 90, 208)-Net over F₈ — Constructive and digital

Digital (49, 90, 208)-net over F₈, using

2 times m-reduction [i] based on digital (49, 92, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 46, 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]

(90−41, 90, 258)-Net over F₈ — Digital

Digital (49, 90, 258)-net over F₈, using

trace code for nets [i] based on digital (4, 45, 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]

(90−41, 90, 12374)-Net in Base 8 — Upper bound on s

There is no (49, 90, 12375)-net in base 8, because

1 times m-reduction [i] would yield (49, 89, 12375)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 237 359527 060220 873608 285080 567735 176274 722572 697790 939021 089452 155035 186377 927051 > 8⁸⁹ [i]