MinT - Best Known (64, 64+29, s)-Nets in Base 8

Best Known (64, 64+29, s)-Nets in Base 8

(64, 64+29, 388)-Net over F₈ — Constructive and digital

Digital (64, 93, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 21, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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 (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]

(64, 64+29, 576)-Net in Base 8 — Constructive

(64, 93, 576)-net in base 8, using

3 times m-reduction [i] based on (64, 96, 576)-net in base 8, using
- trace code for nets [i] based on (16, 48, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
    - base change [i] based on digital (9, 42, 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]

(64, 64+29, 1626)-Net over F₈ — Digital

Digital (64, 93, 1626)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+29, 742926)-Net in Base 8 — Upper bound on s

There is no (64, 93, 742927)-net in base 8, because

1 times m-reduction [i] would yield (64, 92, 742927)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 121417 073457 865881 410125 400268 737423 015434 856585 814573 741486 389987 250528 170483 055153 > 8⁹² [i]