MinT - Best Known (140−45, 140, s)-Nets in Base 8

Best Known (140−45, 140, s)-Nets in Base 8

(140−45, 140, 419)-Net over F₈ — Constructive and digital

Digital (95, 140, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 36, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]
2. digital (59, 104, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 52, 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]

(140−45, 140, 576)-Net in Base 8 — Constructive

(95, 140, 576)-net in base 8, using

10 times m-reduction [i] based on (95, 150, 576)-net in base 8, using
- trace code for nets [i] based on (20, 75, 288)-net in base 64, using
  - 2 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 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]

(140−45, 140, 1864)-Net over F₈ — Digital

Digital (95, 140, 1864)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−45, 140, 657101)-Net in Base 8 — Upper bound on s

There is no (95, 140, 657102)-net in base 8, because

1 times m-reduction [i] would yield (95, 139, 657102)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 338462 076882 561231 805177 295224 927763 806902 910352 521498 207513 269809 007613 049217 268845 042629 613055 450749 368321 273663 541318 164816 > 8¹³⁹ [i]