MinT - Best Known (67, 67+36, s)-Nets in Base 8

Best Known (67, 67+36, s)-Nets in Base 8

(67, 67+36, 354)-Net over F₈ — Constructive and digital

Digital (67, 103, 354)-net over F₈, using

17 times m-reduction [i] based on digital (67, 120, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 60, 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]

(67, 67+36, 518)-Net in Base 8 — Constructive

(67, 103, 518)-net in base 8, using

1 times m-reduction [i] based on (67, 104, 518)-net in base 8, using
- base change [i] based on digital (41, 78, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 39, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(67, 67+36, 921)-Net over F₈ — Digital

Digital (67, 103, 921)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+36, 158738)-Net in Base 8 — Upper bound on s

There is no (67, 103, 158739)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1042 976244 505388 484880 666126 179345 981798 461713 295918 895804 305257 525145 949019 243423 611365 013265 > 8¹⁰³ [i]