MinT - Best Known (137−53, 137, s)-Nets in Base 8

Best Known (137−53, 137, s)-Nets in Base 8

(137−53, 137, 354)-Net over F₈ — Constructive and digital

Digital (84, 137, 354)-net over F₈, using

17 times m-reduction [i] based on digital (84, 154, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 77, 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]

(137−53, 137, 432)-Net in Base 8 — Constructive

(84, 137, 432)-net in base 8, using

1 times m-reduction [i] based on (84, 138, 432)-net in base 8, using
- trace code for nets [i] based on (15, 69, 216)-net in base 64, using
  - 1 times m-reduction [i] based on (15, 70, 216)-net in base 64, using
    - base change [i] based on digital (5, 60, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(137−53, 137, 699)-Net over F₈ — Digital

Digital (84, 137, 699)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−53, 137, 79792)-Net in Base 8 — Upper bound on s

There is no (84, 137, 79793)-net in base 8, because

1 times m-reduction [i] would yield (84, 136, 79793)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 661 163371 946537 982274 180516 957109 227698 598209 098737 703315 315576 579446 586465 729471 271844 455619 282201 306794 924083 381690 573360 > 8¹³⁶ [i]