MinT - Best Known (111−32, 111, s)-Nets in Base 8

Best Known (111−32, 111, s)-Nets in Base 8

(111−32, 111, 484)-Net over F₈ — Constructive and digital

Digital (79, 111, 484)-net over F₈, using

1 times m-reduction [i] based on digital (79, 112, 484)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (16, 32, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 16, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. digital (47, 80, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 40, 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]

(111−32, 111, 576)-Net in Base 8 — Constructive

(79, 111, 576)-net in base 8, using

11 times m-reduction [i] based on (79, 122, 576)-net in base 8, using
- trace code for nets [i] based on (18, 61, 288)-net in base 64, using
  - 2 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(111−32, 111, 3053)-Net over F₈ — Digital

Digital (79, 111, 3053)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−32, 111, 1789060)-Net in Base 8 — Upper bound on s

There is no (79, 111, 1789061)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 17498 041050 880428 132583 467302 883799 538334 661101 801971 638198 890701 713822 165298 347362 303167 180767 783128 > 8¹¹¹ [i]