MinT - Best Known (154−55, 154, s)-Nets in Base 8

Best Known (154−55, 154, s)-Nets in Base 8

(154−55, 154, 378)-Net over F₈ — Constructive and digital

Digital (99, 154, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 30, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [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]

(154−55, 154, 576)-Net in Base 8 — Constructive

(99, 154, 576)-net in base 8, using

t-expansion [i] based on (97, 154, 576)-net in base 8, using
- trace code for nets [i] based on (20, 77, 288)-net in base 64, using
  - base change [i] based on digital (9, 66, 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]

(154−55, 154, 1154)-Net over F₈ — Digital

Digital (99, 154, 1154)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(154−55, 154, 204542)-Net in Base 8 — Upper bound on s

There is no (99, 154, 204543)-net in base 8, because

1 times m-reduction [i] would yield (99, 153, 204543)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 488728 084753 341469 131012 641475 027643 977504 810312 997145 863024 301325 761249 886192 504381 915045 782885 047202 582469 528537 856981 072934 371532 870020 > 8¹⁵³ [i]