MinT - Best Known (166−71, 166, s)-Nets in Base 8

Best Known (166−71, 166, s)-Nets in Base 8

(166−71, 166, 354)-Net over F₈ — Constructive and digital

Digital (95, 166, 354)-net over F₈, using

t-expansion [i] based on digital (93, 166, 354)-net over F₈, using
- 6 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]

(166−71, 166, 514)-Net over F₈ — Digital

Digital (95, 166, 514)-net over F₈, using

trace code for nets [i] based on digital (12, 83, 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]

(166−71, 166, 35918)-Net in Base 8 — Upper bound on s

There is no (95, 166, 35919)-net in base 8, because

1 times m-reduction [i] would yield (95, 165, 35919)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102294 547642 004660 272774 560126 573638 879268 651297 648028 642666 782132 421931 176700 043940 842813 551060 331196 405207 098142 246381 740876 598294 613201 925987 986760 > 8¹⁶⁵ [i]