MinT - Best Known (59−18, 59, s)-Nets in Base 8

Best Known (59−18, 59, s)-Nets in Base 8

(59−18, 59, 363)-Net over F₈ — Constructive and digital

Digital (41, 59, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 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]
  - a shift-net [i]
2. digital (32, 50, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 25, 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]

(59−18, 59, 531)-Net in Base 8 — Constructive

(41, 59, 531)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 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. (30, 48, 514)-net in base 8, using
  - base change [i] based on digital (18, 36, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 18, 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]

(59−18, 59, 1405)-Net over F₈ — Digital

Digital (41, 59, 1405)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−18, 59, 493066)-Net in Base 8 — Upper bound on s

There is no (41, 59, 493067)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 191564 337139 098149 290731 682525 146711 170536 142408 893678 > 8⁵⁹ [i]