MinT - Best Known (82, 137, s)-Nets in Base 8

Best Known (82, 137, s)-Nets in Base 8

(82, 137, 354)-Net over F₈ — Constructive and digital

Digital (82, 137, 354)-net over F₈, using

13 times m-reduction [i] based on digital (82, 150, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 75, 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]

(82, 137, 384)-Net in Base 8 — Constructive

(82, 137, 384)-net in base 8, using

1 times m-reduction [i] based on (82, 138, 384)-net in base 8, using
- trace code for nets [i] based on (13, 69, 192)-net in base 64, using
  - 1 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
    - base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(82, 137, 583)-Net over F₈ — Digital

Digital (82, 137, 583)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 137, 55217)-Net in Base 8 — Upper bound on s

There is no (82, 137, 55218)-net in base 8, because

1 times m-reduction [i] would yield (82, 136, 55218)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 661 334684 575873 437458 321259 068348 800019 854110 211166 791483 842594 179565 589375 003803 392561 212739 311101 316041 829246 535958 103408 > 8¹³⁶ [i]