MinT - Best Known (109−36, 109, s)-Nets in Base 8

Best Known (109−36, 109, s)-Nets in Base 8

(109−36, 109, 382)-Net over F₈ — Constructive and digital

Digital (73, 109, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 23, 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 (50, 86, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 43, 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]

(109−36, 109, 576)-Net in Base 8 — Constructive

(73, 109, 576)-net in base 8, using

3 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
- trace code for nets [i] based on (17, 56, 288)-net in base 64, using
  - base change [i] based on digital (9, 48, 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]

(109−36, 109, 1308)-Net over F₈ — Digital

Digital (73, 109, 1308)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−36, 109, 317488)-Net in Base 8 — Upper bound on s

There is no (73, 109, 317489)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 273 408863 099702 022141 462551 565301 781922 113314 868634 481138 875732 114212 087910 103858 298592 017298 882640 > 8¹⁰⁹ [i]