MinT - Best Known (57, 57+31, s)-Nets in Base 8

Best Known (57, 57+31, s)-Nets in Base 8

(57, 57+31, 354)-Net over F₈ — Constructive and digital

Digital (57, 88, 354)-net over F₈, using

12 times m-reduction [i] based on digital (57, 100, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 50, 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]

(57, 57+31, 518)-Net in Base 8 — Constructive

(57, 88, 518)-net in base 8, using

base change [i] based on digital (35, 66, 518)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 33, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(57, 57+31, 782)-Net over F₈ — Digital

Digital (57, 88, 782)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+31, 158695)-Net in Base 8 — Upper bound on s

There is no (57, 88, 158696)-net in base 8, because

1 times m-reduction [i] would yield (57, 87, 158696)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 705407 595156 214925 679392 214686 830777 205362 904398 229401 142166 347005 340864 481876 > 8⁸⁷ [i]