MinT - Best Known (86, 86+58, s)-Nets in Base 8

Best Known (86, 86+58, s)-Nets in Base 8

(86, 86+58, 354)-Net over F₈ — Constructive and digital

Digital (86, 144, 354)-net over F₈, using

14 times m-reduction [i] based on digital (86, 158, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 79, 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]

(86, 86+58, 384)-Net in Base 8 — Constructive

(86, 144, 384)-net in base 8, using

trace code for nets [i] based on (14, 72, 192)-net in base 64, using
- 5 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
  - base change [i] based on digital (3, 66, 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]

(86, 86+58, 598)-Net over F₈ — Digital

Digital (86, 144, 598)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+58, 50836)-Net in Base 8 — Upper bound on s

There is no (86, 144, 50837)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11092 139611 551957 450058 880404 016472 317062 235241 037932 492504 494526 926931 856542 185455 033405 589243 994645 302781 942266 933525 559713 326848 > 8¹⁴⁴ [i]