MinT - Best Known (99−29, 99, s)-Nets in Base 8

Best Known (99−29, 99, s)-Nets in Base 8

(99−29, 99, 419)-Net over F₈ — Constructive and digital

Digital (70, 99, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (13, 27, 65)-net over F₈, using
  - base reduction for projective spaces (embedding PG(13,64) in PG(26,8)) 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]

(99−29, 99, 576)-Net in Base 8 — Constructive

(70, 99, 576)-net in base 8, using

7 times m-reduction [i] based on (70, 106, 576)-net in base 8, using
- trace code for nets [i] based on (17, 53, 288)-net in base 64, using
  - 3 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]

(99−29, 99, 2531)-Net over F₈ — Digital

Digital (70, 99, 2531)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−29, 99, 1811288)-Net in Base 8 — Upper bound on s

There is no (70, 99, 1811289)-net in base 8, because

1 times m-reduction [i] would yield (70, 98, 1811289)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 31828 871129 101419 770590 517451 495988 961199 166893 053879 662357 393849 191980 818880 135680 384136 > 8⁹⁸ [i]