MinT - Best Known (114−30, 114, s)-Nets in Base 8

Best Known (114−30, 114, s)-Nets in Base 8

(114−30, 114, 610)-Net over F₈ — Constructive and digital

Digital (84, 114, 610)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (25, 40, 256)-net over F₈, using
  - trace code for nets [i] based on digital (5, 20, 128)-net over F₆₄, using
    - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
        a function field by SÃ©mirat [i]
2. digital (44, 74, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 37, 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]

(114−30, 114, 708)-Net in Base 8 — Constructive

(84, 114, 708)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (25, 40, 514)-net in base 8, using
  - base change [i] based on 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]
2. digital (44, 74, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 37, 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]

(114−30, 114, 5931)-Net over F₈ — Digital

Digital (84, 114, 5931)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(114−30, 114, 6701189)-Net in Base 8 — Upper bound on s

There is no (84, 114, 6701190)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8 958985 901784 177607 890632 946178 248876 413972 248648 303939 698915 394605 746284 194896 782955 003675 069699 002816 > 8¹¹⁴ [i]