MinT - Best Known (85, 85+61, s)-Nets in Base 8

Best Known (85, 85+61, s)-Nets in Base 8

(85, 85+61, 354)-Net over F₈ — Constructive and digital

Digital (85, 146, 354)-net over F₈, using

10 times m-reduction [i] based on digital (85, 156, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 78, 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]

(85, 85+61, 514)-Net over F₈ — Digital

Digital (85, 146, 514)-net over F₈, using

trace code for nets [i] based on digital (12, 73, 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]

(85, 85+61, 39849)-Net in Base 8 — Upper bound on s

There is no (85, 146, 39850)-net in base 8, because

1 times m-reduction [i] would yield (85, 145, 39850)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 88770 873334 534227 822902 036488 053895 612754 320818 118311 536371 662345 746366 138678 257879 738507 734080 957379 501522 576831 770672 459494 722064 > 8¹⁴⁵ [i]