MinT - Best Known (168−61, 168, s)-Nets in Base 8

Best Known (168−61, 168, s)-Nets in Base 8

(168−61, 168, 371)-Net over F₈ — Constructive and digital

Digital (107, 168, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 32, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (75, 136, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 68, 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]

(168−61, 168, 576)-Net in Base 8 — Constructive

(107, 168, 576)-net in base 8, using

t-expansion [i] based on (105, 168, 576)-net in base 8, using
- trace code for nets [i] based on (21, 84, 288)-net in base 64, using
  - base change [i] based on digital (9, 72, 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]

(168−61, 168, 1140)-Net over F₈ — Digital

Digital (107, 168, 1140)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−61, 168, 183166)-Net in Base 8 — Upper bound on s

There is no (107, 168, 183167)-net in base 8, because

1 times m-reduction [i] would yield (107, 167, 183167)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 546877 747522 451969 255024 227813 496730 030489 609257 913479 435038 418658 723862 031106 641461 395317 480974 501952 786654 299064 504246 392719 632525 529541 889487 689255 > 8¹⁶⁷ [i]