MinT - Best Known (72−25, 72, s)-Nets in Base 8

Best Known (72−25, 72, s)-Nets in Base 8

(72−25, 72, 354)-Net over F₈ — Constructive and digital

Digital (47, 72, 354)-net over F₈, using

8 times m-reduction [i] based on digital (47, 80, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 40, 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]

(72−25, 72, 518)-Net in Base 8 — Constructive

(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]

(72−25, 72, 729)-Net over F₈ — Digital

Digital (47, 72, 729)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−25, 72, 166543)-Net in Base 8 — Upper bound on s

There is no (47, 72, 166544)-net in base 8, because

1 times m-reduction [i] would yield (47, 71, 166544)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 13164 766285 765717 175411 937473 006087 243460 021592 064465 769070 050359 > 8⁷¹ [i]