MinT - Best Known (123−49, 123, s)-Nets in Base 8

Best Known (123−49, 123, s)-Nets in Base 8

(123−49, 123, 354)-Net over F₈ — Constructive and digital

Digital (74, 123, 354)-net over F₈, using

11 times m-reduction [i] based on digital (74, 134, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 67, 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]

(123−49, 123, 384)-Net in Base 8 — Constructive

(74, 123, 384)-net in base 8, using

1 times m-reduction [i] based on (74, 124, 384)-net in base 8, using
- trace code for nets [i] based on (12, 62, 192)-net in base 64, using
  - 1 times m-reduction [i] based on (12, 63, 192)-net in base 64, using
    - base change [i] based on digital (3, 54, 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]

(123−49, 123, 555)-Net over F₈ — Digital

Digital (74, 123, 555)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−49, 123, 54557)-Net in Base 8 — Upper bound on s

There is no (74, 123, 54558)-net in base 8, because

1 times m-reduction [i] would yield (74, 122, 54558)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 150 349259 060240 692878 122750 374797 793373 242478 209029 880847 108926 285257 798058 120146 786776 666451 736220 523787 407443 > 8¹²² [i]