MinT - Best Known (80, 80+35, s)-Nets in Base 8

Best Known (80, 80+35, s)-Nets in Base 8

(80, 80+35, 419)-Net over F₈ — Constructive and digital

Digital (80, 115, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 31, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [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]

(80, 80+35, 576)-Net in Base 8 — Constructive

(80, 115, 576)-net in base 8, using

9 times m-reduction [i] based on (80, 124, 576)-net in base 8, using
- trace code for nets [i] based on (18, 62, 288)-net in base 64, using
  - 1 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]

(80, 80+35, 2209)-Net over F₈ — Digital

Digital (80, 115, 2209)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(80, 80+35, 1166429)-Net in Base 8 — Upper bound on s

There is no (80, 115, 1166430)-net in base 8, because

1 times m-reduction [i] would yield (80, 114, 1166430)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8 959045 645334 002107 294761 946768 972851 612930 325404 488613 613099 845050 674131 534968 458601 034828 101466 159117 > 8¹¹⁴ [i]