MinT - Best Known (74−25, 74, s)-Nets in Base 8

Best Known (74−25, 74, s)-Nets in Base 8

(74−25, 74, 354)-Net over F₈ — Constructive and digital

Digital (49, 74, 354)-net over F₈, using

10 times m-reduction [i] based on digital (49, 84, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 42, 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]

(74−25, 74, 518)-Net in Base 8 — Constructive

(49, 74, 518)-net in base 8, using

8² times duplication [i] based on (47, 72, 518)-net in base 8, using
- base change [i] based on digital (29, 54, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 27, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(74−25, 74, 865)-Net over F₈ — Digital

Digital (49, 74, 865)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(74−25, 74, 235530)-Net in Base 8 — Upper bound on s

There is no (49, 74, 235531)-net in base 8, because

1 times m-reduction [i] would yield (49, 73, 235531)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 842511 916888 132870 223345 646144 489768 309600 411245 231034 449480 229504 > 8⁷³ [i]