MinT - Best Known (165−65, 165, s)-Nets in Base 8

Best Known (165−65, 165, s)-Nets in Base 8

(165−65, 165, 354)-Net over F₈ — Constructive and digital

Digital (100, 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]

(165−65, 165, 432)-Net in Base 8 — Constructive

(100, 165, 432)-net in base 8, using

1 times m-reduction [i] based on (100, 166, 432)-net in base 8, using
- trace code for nets [i] based on (17, 83, 216)-net in base 64, using
  - 1 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]

(165−65, 165, 745)-Net over F₈ — Digital

Digital (100, 165, 745)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−65, 165, 77625)-Net in Base 8 — Upper bound on s

There is no (100, 165, 77626)-net in base 8, because

1 times m-reduction [i] would yield (100, 164, 77626)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12787 593904 955790 263932 282977 638979 838452 483692 035518 988054 072707 541679 005631 618379 441412 600843 717777 761492 265926 777323 664705 436007 005131 549474 319731 > 8¹⁶⁴ [i]