MinT - Best Known (101, 101+53, s)-Nets in Base 8

Best Known (101, 101+53, s)-Nets in Base 8

(101, 101+53, 389)-Net over F₈ — Constructive and digital

Digital (101, 154, 389)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 34, 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 (67, 120, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 60, 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, 101+53, 576)-Net in Base 8 — Constructive

(101, 154, 576)-net in base 8, using

6 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, 101+53, 1392)-Net over F₈ — Digital

Digital (101, 154, 1392)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(101, 101+53, 310820)-Net in Base 8 — Upper bound on s

There is no (101, 154, 310821)-net in base 8, because

1 times m-reduction [i] would yield (101, 153, 310821)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 488634 968596 741261 772103 958157 632651 597474 745454 920639 198909 765249 733378 177832 233178 002080 016122 769035 236514 111404 440961 471707 173779 519336 > 8¹⁵³ [i]