MinT - Best Known (159−61, 159, s)-Nets in Base 8

Best Known (159−61, 159, s)-Nets in Base 8

(159−61, 159, 354)-Net over F₈ — Constructive and digital

Digital (98, 159, 354)-net over F₈, using

t-expansion [i] based on digital (93, 159, 354)-net over F₈, using
- 13 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(159−61, 159, 432)-Net in Base 8 — Constructive

(98, 159, 432)-net in base 8, using

3 times m-reduction [i] based on (98, 162, 432)-net in base 8, using
- trace code for nets [i] based on (17, 81, 216)-net in base 64, using
  - 3 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 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]

(159−61, 159, 822)-Net over F₈ — Digital

Digital (98, 159, 822)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(159−61, 159, 98147)-Net in Base 8 — Upper bound on s

There is no (98, 159, 98148)-net in base 8, because

1 times m-reduction [i] would yield (98, 158, 98148)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 48777 339788 283589 622126 506993 431507 326142 509757 049264 031757 054932 161735 804192 439634 937049 722812 584544 463643 846605 942357 574123 767871 962398 978728 > 8¹⁵⁸ [i]