MinT - Best Known (69, 69+45, s)-Nets in Base 8

Best Known (69, 69+45, s)-Nets in Base 8

(69, 69+45, 354)-Net over F₈ — Constructive and digital

Digital (69, 114, 354)-net over F₈, using

10 times m-reduction [i] based on digital (69, 124, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 62, 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]

(69, 69+45, 384)-Net in Base 8 — Constructive

(69, 114, 384)-net in base 8, using

8² times duplication [i] based on (67, 112, 384)-net in base 8, using
- trace code for nets [i] based on (11, 56, 192)-net in base 64, using
  - base change [i] based on digital (3, 48, 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]

(69, 69+45, 547)-Net over F₈ — Digital

Digital (69, 114, 547)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+45, 56266)-Net in Base 8 — Upper bound on s

There is no (69, 114, 56267)-net in base 8, because

1 times m-reduction [i] would yield (69, 113, 56267)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 120183 577593 148862 973530 043769 392315 770514 453476 134137 581849 978230 104480 185573 809861 843643 509953 659528 > 8¹¹³ [i]