MinT - Best Known (162−69, 162, s)-Nets in Base 8

Best Known (162−69, 162, s)-Nets in Base 8

(162−69, 162, 354)-Net over F₈ — Constructive and digital

Digital (93, 162, 354)-net over F₈, using

10 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 86, 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]

(162−69, 162, 514)-Net over F₈ — Digital

Digital (93, 162, 514)-net over F₈, using

trace code for nets [i] based on digital (12, 81, 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]

(162−69, 162, 36519)-Net in Base 8 — Upper bound on s

There is no (93, 162, 36520)-net in base 8, because

1 times m-reduction [i] would yield (93, 161, 36520)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 24 994594 363158 374093 063647 225939 490209 177980 262381 344082 499374 753345 931557 528716 684504 761520 453382 291390 577868 539004 231559 340533 242250 599175 910297 > 8¹⁶¹ [i]