MinT - Best Known (81, 81+53, s)-Nets in Base 8

Best Known (81, 81+53, s)-Nets in Base 8

(81, 81+53, 354)-Net over F₈ — Constructive and digital

Digital (81, 134, 354)-net over F₈, using

14 times m-reduction [i] based on digital (81, 148, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 74, 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]

(81, 81+53, 384)-Net in Base 8 — Constructive

(81, 134, 384)-net in base 8, using

2 times m-reduction [i] based on (81, 136, 384)-net in base 8, using
- trace code for nets [i] based on (13, 68, 192)-net in base 64, using
  - 2 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
    - base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(81, 81+53, 615)-Net over F₈ — Digital

Digital (81, 134, 615)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+53, 62767)-Net in Base 8 — Upper bound on s

There is no (81, 134, 62768)-net in base 8, because

1 times m-reduction [i] would yield (81, 133, 62768)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 291343 662647 760509 670589 349031 465319 589516 158111 249034 348273 800116 663480 128627 770294 667893 997666 710754 149196 793552 920206 > 8¹³³ [i]