MinT - Best Known (112−33, 112, s)-Nets in Base 8

Best Known (112−33, 112, s)-Nets in Base 8

(112−33, 112, 484)-Net over F₈ — Constructive and digital

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]

(112−33, 112, 576)-Net in Base 8 — Constructive

(79, 112, 576)-net in base 8, using

10 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]

(112−33, 112, 2662)-Net over F₈ — Digital

Digital (79, 112, 2662)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−33, 112, 1789060)-Net in Base 8 — Upper bound on s

There is no (79, 112, 1789061)-net in base 8, because

1 times m-reduction [i] would yield (79, 111, 1789061)-net in base 8, but
- 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]