MinT - Best Known (57−19, 57, s)-Nets in Base 8

Best Known (57−19, 57, s)-Nets in Base 8

(57−19, 57, 354)-Net over F₈ — Constructive and digital

Digital (38, 57, 354)-net over F₈, using

5 times m-reduction [i] based on digital (38, 62, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 31, 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]

(57−19, 57, 518)-Net in Base 8 — Constructive

(38, 57, 518)-net in base 8, using

8¹ times duplication [i] based on (37, 56, 518)-net in base 8, using
- base change [i] based on digital (23, 42, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 21, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(57−19, 57, 790)-Net over F₈ — Digital

Digital (38, 57, 790)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−19, 57, 246530)-Net in Base 8 — Upper bound on s

There is no (38, 57, 246531)-net in base 8, because

1 times m-reduction [i] would yield (38, 56, 246531)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 374 151046 827084 537480 270470 882813 870968 811852 401600 > 8⁵⁶ [i]