MinT - Best Known (138−56, 138, s)-Nets in Base 8

Best Known (138−56, 138, s)-Nets in Base 8

(138−56, 138, 354)-Net over F₈ — Constructive and digital

Digital (82, 138, 354)-net over F₈, using

12 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]

(138−56, 138, 384)-Net in Base 8 — Constructive

(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]

(138−56, 138, 558)-Net over F₈ — Digital

Digital (82, 138, 558)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−56, 138, 45570)-Net in Base 8 — Upper bound on s

There is no (82, 138, 45571)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 42320 639873 144785 800963 171292 744934 702149 450038 139736 967348 177680 178882 440712 389033 795971 368782 586647 567781 736231 398547 914624 > 8¹³⁸ [i]