MinT - Best Known (169−81, 169, s)-Nets in Base 8

Best Known (169−81, 169, s)-Nets in Base 8

(169−81, 169, 208)-Net over F₈ — Constructive and digital

Digital (88, 169, 208)-net over F₈, using

1 times m-reduction [i] based on digital (88, 170, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 85, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(169−81, 169, 225)-Net in Base 8 — Constructive

(88, 169, 225)-net in base 8, using

t-expansion [i] based on (83, 169, 225)-net in base 8, using
- 3 times m-reduction [i] based on (83, 172, 225)-net in base 8, using
  - base change [i] based on digital (40, 129, 225)-net over F₁₆, using
    - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
        a function field by GarcÃa and Quoos [i]

(169−81, 169, 322)-Net over F₈ — Digital

Digital (88, 169, 322)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−81, 169, 13960)-Net in Base 8 — Upper bound on s

There is no (88, 169, 13961)-net in base 8, because

1 times m-reduction [i] would yield (88, 168, 13961)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 512952 213063 595863 598927 295881 730097 364715 313027 572499 582332 693181 789178 448066 340964 454652 543438 993948 674056 806761 996439 977573 981304 814692 918416 675804 > 8¹⁶⁸ [i]