MinT - Best Known (159−55, 159, s)-Nets in Base 8

Best Known (159−55, 159, s)-Nets in Base 8

(159−55, 159, 389)-Net over F₈ — Constructive and digital

Digital (104, 159, 389)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 35, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using a function field by SÃ©mirat [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]

(159−55, 159, 576)-Net in Base 8 — Constructive

(104, 159, 576)-net in base 8, using

7 times m-reduction [i] based on (104, 166, 576)-net in base 8, using
- trace code for nets [i] based on (21, 83, 288)-net in base 64, using
  - 1 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]

(159−55, 159, 1393)-Net over F₈ — Digital

Digital (104, 159, 1393)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(159−55, 159, 300630)-Net in Base 8 — Upper bound on s

There is no (104, 159, 300631)-net in base 8, because

1 times m-reduction [i] would yield (104, 158, 300631)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 48779 048317 056986 563933 502093 385997 712884 889129 376018 444785 404137 529918 638283 262643 735851 602049 443818 370641 737980 701795 915767 744254 362552 328992 > 8¹⁵⁸ [i]