MinT - Best Known (89, 89+65, s)-Nets in Base 8

Best Known (89, 89+65, s)-Nets in Base 8

(89, 89+65, 354)-Net over F₈ — Constructive and digital

Digital (89, 154, 354)-net over F₈, using

10 times m-reduction [i] based on digital (89, 164, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 82, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(89, 89+65, 514)-Net over F₈ — Digital

Digital (89, 154, 514)-net over F₈, using

trace code for nets [i] based on digital (12, 77, 257)-net over F₆₄, using
- net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(89, 89+65, 37970)-Net in Base 8 — Upper bound on s

There is no (89, 154, 37971)-net in base 8, because

1 times m-reduction [i] would yield (89, 153, 37971)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 488585 140745 332159 691302 206175 853900 501920 926004 823184 190754 491380 530046 754446 717449 449903 448808 668808 989456 643597 089856 461862 642760 990866 > 8¹⁵³ [i]