MinT - Best Known (51, 51+22, s)-Nets in Base 8

Best Known (51, 51+22, s)-Nets in Base 8

(51, 51+22, 379)-Net over F₈ — Constructive and digital

Digital (51, 73, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 15, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (36, 58, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 29, 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]

(51, 51+22, 531)-Net in Base 8 — Constructive

(51, 73, 531)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. (38, 60, 514)-net in base 8, using
  - base change [i] based on digital (23, 45, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (23, 46, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 23, 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]

(51, 51+22, 1719)-Net over F₈ — Digital

Digital (51, 73, 1719)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+22, 690471)-Net in Base 8 — Upper bound on s

There is no (51, 73, 690472)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 842509 831923 590891 737527 004860 818020 899131 666530 455693 287366 977460 > 8⁷³ [i]