MinT - Best Known (88, 88+45, s)-Nets in Base 8

Best Known (88, 88+45, s)-Nets in Base 8

(88, 88+45, 388)-Net over F₈ — Constructive and digital

Digital (88, 133, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 29, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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 (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]

(88, 88+45, 576)-Net in Base 8 — Constructive

(88, 133, 576)-net in base 8, using

5 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]

(88, 88+45, 1345)-Net over F₈ — Digital

Digital (88, 133, 1345)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(88, 88+45, 339060)-Net in Base 8 — Upper bound on s

There is no (88, 133, 339061)-net in base 8, because

1 times m-reduction [i] would yield (88, 132, 339061)-net in base 8, but
- 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]