MinT - Best Known (89, 89+51, s)-Nets in Base 8

Best Known (89, 89+51, s)-Nets in Base 8

(89, 89+51, 354)-Net over F₈ — Constructive and digital

Digital (89, 140, 354)-net over F₈, using

24 times m-reduction [i] based on digital (89, 164, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 82, 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]

(89, 89+51, 576)-Net in Base 8 — Constructive

(89, 140, 576)-net in base 8, using

trace code for nets [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]

(89, 89+51, 965)-Net over F₈ — Digital

Digital (89, 140, 965)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+51, 152638)-Net in Base 8 — Upper bound on s

There is no (89, 140, 152639)-net in base 8, because

1 times m-reduction [i] would yield (89, 139, 152639)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 338493 742319 368897 295968 302176 155861 696706 211026 003821 826946 056589 659078 985679 292226 045285 825758 373838 561284 514872 114499 465578 > 8¹³⁹ [i]