MinT - Best Known (57, 57+32, s)-Nets in Base 8

Best Known (57, 57+32, s)-Nets in Base 8

(57, 57+32, 354)-Net over F₈ — Constructive and digital

Digital (57, 89, 354)-net over F₈, using

11 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+32, 516)-Net in Base 8 — Constructive

(57, 89, 516)-net in base 8, using

8¹ times duplication [i] based on (56, 88, 516)-net in base 8, using
- base change [i] based on digital (34, 66, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 33, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(57, 57+32, 710)-Net over F₈ — Digital

Digital (57, 89, 710)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+32, 102526)-Net in Base 8 — Upper bound on s

There is no (57, 89, 102527)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 237 150592 417990 503491 660738 166473 361571 176224 139612 672926 575520 295499 339762 243065 > 8⁸⁹ [i]