MinT - Best Known (86−26, 86, s)-Nets in Base 8

Best Known (86−26, 86, s)-Nets in Base 8

(86−26, 86, 388)-Net over F₈ — Constructive and digital

Digital (60, 86, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 20, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (40, 66, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 33, 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]

(86−26, 86, 576)-Net in Base 8 — Constructive

(60, 86, 576)-net in base 8, using

2 times m-reduction [i] based on (60, 88, 576)-net in base 8, using
- trace code for nets [i] based on (16, 44, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
    - base change [i] based on digital (9, 42, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(86−26, 86, 1871)-Net over F₈ — Digital

Digital (60, 86, 1871)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−26, 86, 763117)-Net in Base 8 — Upper bound on s

There is no (60, 86, 763118)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 463174 244462 328285 048610 864417 291299 189619 270022 552171 593426 598331 298158 494307 > 8⁸⁶ [i]