MinT - Best Known (172−83, 172, s)-Nets in Base 8

Best Known (172−83, 172, s)-Nets in Base 8

(172−83, 172, 208)-Net over F₈ — Constructive and digital

Digital (89, 172, 208)-net over F₈, using

trace code for nets [i] based on digital (3, 86, 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]

(172−83, 172, 225)-Net in Base 8 — Constructive

(89, 172, 225)-net in base 8, using

t-expansion [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]

(172−83, 172, 317)-Net over F₈ — Digital

Digital (89, 172, 317)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−83, 172, 13443)-Net in Base 8 — Upper bound on s

There is no (89, 172, 13444)-net in base 8, because

1 times m-reduction [i] would yield (89, 171, 13444)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26819 662200 317660 913073 249926 414395 531798 571454 366263 089134 465480 346191 852085 434733 459779 272990 251216 247691 592912 997255 045082 948553 492645 532889 791604 462960 > 8¹⁷¹ [i]