MinT - Best Known (172−51, 172, s)-Nets in Base 8

Best Known (172−51, 172, s)-Nets in Base 8

(172−51, 172, 1026)-Net over F₈ — Constructive and digital

Digital (121, 172, 1026)-net over F₈, using

t-expansion [i] based on digital (114, 172, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 86, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(172−51, 172, 3583)-Net over F₈ — Digital

Digital (121, 172, 3583)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−51, 172, 2186050)-Net in Base 8 — Upper bound on s

There is no (121, 172, 2186051)-net in base 8, because

1 times m-reduction [i] would yield (121, 171, 2186051)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26815 747136 544713 041461 286742 192646 549060 129529 662827 644609 640157 411624 142505 574441 400530 356928 192885 272851 799869 280759 172762 801850 929088 013265 387989 926512 > 8¹⁷¹ [i]