MinT - Best Known (111−40, 111, s)-Nets in Base 8

Best Known (111−40, 111, s)-Nets in Base 8

(111−40, 111, 354)-Net over F₈ — Constructive and digital

Digital (71, 111, 354)-net over F₈, using

17 times m-reduction [i] based on digital (71, 128, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 64, 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]

(111−40, 111, 516)-Net in Base 8 — Constructive

(71, 111, 516)-net in base 8, using

1 times m-reduction [i] based on (71, 112, 516)-net in base 8, using
- base change [i] based on digital (43, 84, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 42, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(111−40, 111, 837)-Net over F₈ — Digital

Digital (71, 111, 837)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−40, 111, 121987)-Net in Base 8 — Upper bound on s

There is no (71, 111, 121988)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 17500 811865 269529 729775 655938 668384 464312 771775 705084 439850 013841 908658 550371 107231 062636 137747 258784 > 8¹¹¹ [i]