MinT - Best Known (163−55, 163, s)-Nets in Base 8

Best Known (163−55, 163, s)-Nets in Base 8

(163−55, 163, 402)-Net over F₈ — Constructive and digital

Digital (108, 163, 402)-net over F₈, using

8¹ times duplication [i] based on digital (107, 162, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 38, 48)-net over F₈, using
    - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
        a function field by SÃ©mirat [i]
  2. 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]

(163−55, 163, 576)-Net in Base 8 — Constructive

(108, 163, 576)-net in base 8, using

9 times m-reduction [i] based on (108, 172, 576)-net in base 8, using
- trace code for nets [i] based on (22, 86, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
    - base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(163−55, 163, 1621)-Net over F₈ — Digital

Digital (108, 163, 1621)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(163−55, 163, 409101)-Net in Base 8 — Upper bound on s

There is no (108, 163, 409102)-net in base 8, because

1 times m-reduction [i] would yield (108, 162, 409102)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 199 802406 976152 029611 825418 657938 865391 121774 339183 871530 472509 153916 825753 149064 164025 817779 782822 509558 237886 240961 635650 073486 842189 855004 679616 > 8¹⁶² [i]