MinT - Best Known (165−66, 165, s)-Nets in Base 8

Best Known (165−66, 165, s)-Nets in Base 8

(165−66, 165, 354)-Net over F₈ — Constructive and digital

Digital (99, 165, 354)-net over F₈, using

t-expansion [i] based on digital (93, 165, 354)-net over F₈, using
- 7 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]

(165−66, 165, 384)-Net in Base 8 — Constructive

(99, 165, 384)-net in base 8, using

3 times m-reduction [i] based on (99, 168, 384)-net in base 8, using
- trace code for nets [i] based on (15, 84, 192)-net in base 64, using
  - base change [i] based on digital (3, 72, 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]

(165−66, 165, 693)-Net over F₈ — Digital

Digital (99, 165, 693)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−66, 165, 61597)-Net in Base 8 — Upper bound on s

There is no (99, 165, 61598)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102344 851695 798929 793837 955490 789888 322137 072635 638758 211972 471158 438122 131763 995878 302941 679317 339246 180469 965739 411761 945543 017258 107649 569663 720044 > 8¹⁶⁵ [i]