MinT - Best Known (87−28, 87, s)-Nets in Base 8

Best Known (87−28, 87, s)-Nets in Base 8

(87−28, 87, 378)-Net over F₈ — Constructive and digital

Digital (59, 87, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 17, 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 (42, 70, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 35, 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]

(87−28, 87, 522)-Net in Base 8 — Constructive

(59, 87, 522)-net in base 8, using

1 times m-reduction [i] based on (59, 88, 522)-net in base 8, using
- base change [i] based on digital (37, 66, 522)-net over F₁₆, using
  - trace code for nets [i] based on digital (4, 33, 261)-net over F₂₅₆, using
    - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(87−28, 87, 1282)-Net over F₈ — Digital

Digital (59, 87, 1282)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−28, 87, 353515)-Net in Base 8 — Upper bound on s

There is no (59, 87, 353516)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 705419 996844 497680 310966 462409 923508 535389 218861 633474 348160 557654 770638 536472 > 8⁸⁷ [i]