MinT - Best Known (52−15, 52, s)-Nets in Base 8

Best Known (52−15, 52, s)-Nets in Base 8

(52−15, 52, 368)-Net over F₈ — Constructive and digital

Digital (37, 52, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 16, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 8, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (21, 36, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 18, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(52−15, 52, 547)-Net in Base 8 — Constructive

(37, 52, 547)-net in base 8, using

base change [i] based on digital (24, 39, 547)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (2, 9, 33)-net over F₁₆, using
    - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
  2. digital (15, 30, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 15, 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]

(52−15, 52, 1960)-Net over F₈ — Digital

Digital (37, 52, 1960)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(52−15, 52, 1834320)-Net in Base 8 — Upper bound on s

There is no (37, 52, 1834321)-net in base 8, because

1 times m-reduction [i] would yield (37, 51, 1834321)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11417 992130 085948 527952 121730 262797 810433 267664 > 8⁵¹ [i]