MinT - Best Known (132−44, 132, s)-Nets in Base 8

Best Known (132−44, 132, s)-Nets in Base 8

(132−44, 132, 389)-Net over F₈ — Constructive and digital

Digital (88, 132, 389)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 30, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using a function field by SÃ©mirat [i]
2. digital (58, 102, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 51, 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]

(132−44, 132, 576)-Net in Base 8 — Constructive

(88, 132, 576)-net in base 8, using

6 times m-reduction [i] based on (88, 138, 576)-net in base 8, using
- trace code for nets [i] based on (19, 69, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(132−44, 132, 1449)-Net over F₈ — Digital

Digital (88, 132, 1449)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(132−44, 132, 339060)-Net in Base 8 — Upper bound on s

There is no (88, 132, 339061)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 161393 919556 279378 198610 566156 073072 840525 765830 626465 858048 794235 474219 294720 945704 400331 258617 063854 393313 183639 771216 > 8¹³² [i]