MinT - Best Known (91−29, 91, s)-Nets in Base 8

Best Known (91−29, 91, s)-Nets in Base 8

(91−29, 91, 382)-Net over F₈ — Constructive and digital

Digital (62, 91, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 19, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (43, 72, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 36, 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]

(91−29, 91, 576)-Net in Base 8 — Constructive

(62, 91, 576)-net in base 8, using

1 times m-reduction [i] based on (62, 92, 576)-net in base 8, using
- trace code for nets [i] based on (16, 46, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
    - base change [i] based on digital (9, 42, 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]

(91−29, 91, 1404)-Net over F₈ — Digital

Digital (62, 91, 1404)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(91−29, 91, 551990)-Net in Base 8 — Upper bound on s

There is no (62, 91, 551991)-net in base 8, because

1 times m-reduction [i] would yield (62, 90, 551991)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1897 167990 531145 760701 574531 833791 102192 107352 108571 072973 057675 631030 473458 617262 > 8⁹⁰ [i]