MinT - Best Known (112−38, 112, s)-Nets in Base 8

Best Known (112−38, 112, s)-Nets in Base 8

(112−38, 112, 378)-Net over F₈ — Constructive and digital

Digital (74, 112, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 22, 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 (52, 90, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 45, 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]

(112−38, 112, 576)-Net in Base 8 — Constructive

(74, 112, 576)-net in base 8, using

t-expansion [i] based on (73, 112, 576)-net in base 8, using
- trace code for nets [i] based on (17, 56, 288)-net in base 64, using
  - base change [i] based on digital (9, 48, 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]

(112−38, 112, 1152)-Net over F₈ — Digital

Digital (74, 112, 1152)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−38, 112, 238546)-Net in Base 8 — Upper bound on s

There is no (74, 112, 238547)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 139984 050860 459496 519862 053008 906209 741292 951282 285008 362594 981960 926213 680110 093585 866743 767201 151032 > 8¹¹² [i]