MinT - Best Known (151−55, 151, s)-Nets in Base 8

Best Known (151−55, 151, s)-Nets in Base 8

(151−55, 151, 363)-Net over F₈ — Constructive and digital

Digital (96, 151, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 27, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (69, 124, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 62, 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]

(151−55, 151, 576)-Net in Base 8 — Constructive

(96, 151, 576)-net in base 8, using

1 times m-reduction [i] based on (96, 152, 576)-net in base 8, using
- trace code for nets [i] based on (20, 76, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 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]

(151−55, 151, 1028)-Net over F₈ — Digital

Digital (96, 151, 1028)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(151−55, 151, 162341)-Net in Base 8 — Upper bound on s

There is no (96, 151, 162342)-net in base 8, because

1 times m-reduction [i] would yield (96, 150, 162342)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2907 474150 483928 009670 436076 190197 495793 427410 811973 418614 622189 040675 092662 524381 820036 255335 317987 619519 708826 914248 512977 833259 763304 > 8¹⁵⁰ [i]