MinT - Best Known (101, 159, s)-Nets in Base 8

Best Known (101, 159, s)-Nets in Base 8

(101, 159, 363)-Net over F₈ — Constructive and digital

Digital (101, 159, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 29, 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 (72, 130, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 65, 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]

(101, 159, 576)-Net in Base 8 — Constructive

(101, 159, 576)-net in base 8, using

1 times m-reduction [i] based on (101, 160, 576)-net in base 8, using
- trace code for nets [i] based on (21, 80, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(101, 159, 1064)-Net over F₈ — Digital

Digital (101, 159, 1064)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(101, 159, 149071)-Net in Base 8 — Upper bound on s

There is no (101, 159, 149072)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 390232 358213 266055 862446 645606 674114 862137 833966 389230 282261 168414 797678 399689 362877 724847 229047 936160 535940 798479 797113 569886 878074 727582 067884 > 8¹⁵⁹ [i]