MinT - Best Known (96−31, 96, s)-Nets in Base 8

Best Known (96−31, 96, s)-Nets in Base 8

(96−31, 96, 382)-Net over F₈ — Constructive and digital

Digital (65, 96, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 20, 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 (45, 76, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 38, 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]

(96−31, 96, 576)-Net in Base 8 — Constructive

(65, 96, 576)-net in base 8, using

2 times m-reduction [i] based on (65, 98, 576)-net in base 8, using
- trace code for nets [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]

(96−31, 96, 1350)-Net over F₈ — Digital

Digital (65, 96, 1350)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(96−31, 96, 481094)-Net in Base 8 — Upper bound on s

There is no (65, 96, 481095)-net in base 8, because

1 times m-reduction [i] would yield (65, 95, 481095)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 62 166646 238052 498691 588150 639281 764020 035626 504149 832380 426674 178067 259024 907433 909120 > 8⁹⁵ [i]