MinT - Best Known (109−35, 109, s)-Nets in Base 8

Best Known (109−35, 109, s)-Nets in Base 8

(109−35, 109, 389)-Net over F₈ — Constructive and digital

Digital (74, 109, 389)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 25, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using a function field by SÃ©mirat [i]
2. digital (49, 84, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 42, 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]

(109−35, 109, 576)-Net in Base 8 — Constructive

(74, 109, 576)-net in base 8, using

t-expansion [i] based on (73, 109, 576)-net in base 8, using
- 3 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
  - trace code for nets [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]

(109−35, 109, 1536)-Net over F₈ — Digital

Digital (74, 109, 1536)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−35, 109, 559907)-Net in Base 8 — Upper bound on s

There is no (74, 109, 559908)-net in base 8, because

1 times m-reduction [i] would yield (74, 108, 559908)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 175833 269248 558241 576092 824025 211211 745149 592170 986123 030398 861916 429930 679951 200754 021549 486382 > 8¹⁰⁸ [i]