MinT - Best Known (100−29, 100, s)-Nets in Base 8

Best Known (100−29, 100, s)-Nets in Base 8

(100−29, 100, 484)-Net over F₈ — Constructive and digital

Digital (71, 100, 484)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 28, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 14, 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 (43, 72, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 36, 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]

(100−29, 100, 576)-Net in Base 8 — Constructive

(71, 100, 576)-net in base 8, using

8 times m-reduction [i] based on (71, 108, 576)-net in base 8, using
- trace code for nets [i] based on (17, 54, 288)-net in base 64, using
  - 2 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 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]

(100−29, 100, 2725)-Net over F₈ — Digital

Digital (71, 100, 2725)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(100−29, 100, 2101330)-Net in Base 8 — Upper bound on s

There is no (71, 100, 2101331)-net in base 8, because

1 times m-reduction [i] would yield (71, 99, 2101331)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 254631 042277 969948 177354 492162 559837 665381 574856 747593 634495 227891 131906 865679 841935 583076 > 8⁹⁹ [i]