MinT - Best Known (64−20, 64, s)-Nets in Base 8

Best Known (64−20, 64, s)-Nets in Base 8

(64−20, 64, 363)-Net over F₈ — Constructive and digital

Digital (44, 64, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (34, 54, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 27, 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]

(64−20, 64, 523)-Net in Base 8 — Constructive

(44, 64, 523)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. (34, 54, 514)-net in base 8, using
  - trace code for nets [i] based on (7, 27, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
      - base change [i] based on digital (0, 21, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(64−20, 64, 1257)-Net over F₈ — Digital

Digital (44, 64, 1257)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(64−20, 64, 389625)-Net in Base 8 — Upper bound on s

There is no (44, 64, 389626)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6277 153490 257261 956728 278731 549135 336098 885186 892187 948824 > 8⁶⁴ [i]