MinT - Best Known (167−59, 167, s)-Nets in Base 8

Best Known (167−59, 167, s)-Nets in Base 8

(167−59, 167, 382)-Net over F₈ — Constructive and digital

Digital (108, 167, 382)-net over F₈, using

8¹ times duplication [i] based on digital (107, 166, 382)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 34, 28)-net over F₈, using
    - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
        a function field by SÃ©mirat [i]
  2. digital (73, 132, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 66, 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]

(167−59, 167, 576)-Net in Base 8 — Constructive

(108, 167, 576)-net in base 8, using

5 times m-reduction [i] based on (108, 172, 576)-net in base 8, using
- trace code for nets [i] based on (22, 86, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
    - base change [i] based on digital (9, 78, 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]

(167−59, 167, 1307)-Net over F₈ — Digital

Digital (108, 167, 1307)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−59, 167, 246265)-Net in Base 8 — Upper bound on s

There is no (108, 167, 246266)-net in base 8, because

1 times m-reduction [i] would yield (108, 166, 246266)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 818392 012635 297565 813147 823330 924757 659278 637245 956415 458537 360300 764716 307810 115999 754149 671192 462648 607912 587330 103234 022909 947240 910436 915220 180088 > 8¹⁶⁶ [i]