MinT - Best Known (172−62, 172, s)-Nets in Base 8

Best Known (172−62, 172, s)-Nets in Base 8

(172−62, 172, 378)-Net over F₈ — Constructive and digital

Digital (110, 172, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 34, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (76, 138, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 69, 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]

(172−62, 172, 576)-Net in Base 8 — Constructive

(110, 172, 576)-net in base 8, using

t-expansion [i] based on (108, 172, 576)-net in base 8, using
- trace code for nets [i] based on (22, 86, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
    - base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(172−62, 172, 1208)-Net over F₈ — Digital

Digital (110, 172, 1208)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−62, 172, 181794)-Net in Base 8 — Upper bound on s

There is no (110, 172, 181795)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 214533 999778 530205 604331 010406 145523 573197 806276 520240 045278 695929 496246 121266 224032 791011 068522 689748 981249 097434 142400 440704 952351 930118 445758 989734 214240 > 8¹⁷² [i]