MinT - Best Known (39, 39+19, s)-Nets in Base 8

Best Known (39, 39+19, s)-Nets in Base 8

(39, 39+19, 354)-Net over F₈ — Constructive and digital

Digital (39, 58, 354)-net over F₈, using

6 times m-reduction [i] based on digital (39, 64, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 32, 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]

(39, 39+19, 518)-Net in Base 8 — Constructive

(39, 58, 518)-net in base 8, using

8² 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]

(39, 39+19, 886)-Net over F₈ — Digital

Digital (39, 58, 886)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(39, 39+19, 310610)-Net in Base 8 — Upper bound on s

There is no (39, 58, 310611)-net in base 8, because

1 times m-reduction [i] would yield (39, 57, 310611)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2993 210958 860632 412527 544219 697724 633505 189977 764212 > 8⁵⁷ [i]