MinT - Best Known (156−53, 156, s)-Nets in Base 8

Best Known (156−53, 156, s)-Nets in Base 8

(156−53, 156, 400)-Net over F₈ — Constructive and digital

Digital (103, 156, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 36, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [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]

(156−53, 156, 576)-Net in Base 8 — Constructive

(103, 156, 576)-net in base 8, using

8 times m-reduction [i] based on (103, 164, 576)-net in base 8, using
- trace code for nets [i] based on (21, 82, 288)-net in base 64, using
  - 2 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]

(156−53, 156, 1505)-Net over F₈ — Digital

Digital (103, 156, 1505)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(156−53, 156, 364738)-Net in Base 8 — Upper bound on s

There is no (103, 156, 364739)-net in base 8, because

1 times m-reduction [i] would yield (103, 155, 364739)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 95 272868 258665 545757 890973 876583 442112 502977 824482 356390 475635 997766 582997 749378 029331 502079 912574 094258 728959 621306 323863 202094 713629 748024 > 8¹⁵⁵ [i]