MinT - Best Known (165−61, 165, s)-Nets in Base 8

Best Known (165−61, 165, s)-Nets in Base 8

(165−61, 165, 354)-Net over F₈ — Constructive and digital

Digital (104, 165, 354)-net over F₈, using

t-expansion [i] based on digital (93, 165, 354)-net over F₈, using
- 7 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(165−61, 165, 576)-Net in Base 8 — Constructive

(104, 165, 576)-net in base 8, using

1 times m-reduction [i] based on (104, 166, 576)-net in base 8, using
- trace code for nets [i] based on (21, 83, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(165−61, 165, 1023)-Net over F₈ — Digital

Digital (104, 165, 1023)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−61, 165, 148774)-Net in Base 8 — Upper bound on s

There is no (104, 165, 148775)-net in base 8, because

1 times m-reduction [i] would yield (104, 164, 148775)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12788 654885 245204 057651 628462 640571 678868 365009 168812 787900 779152 835416 917519 471093 151292 494588 410415 797892 637358 099792 193050 861954 189598 620393 837668 > 8¹⁶⁴ [i]