MinT - Best Known (135−48, 135, s)-Nets in Base 8

Best Known (135−48, 135, s)-Nets in Base 8

(135−48, 135, 368)-Net over F₈ — Constructive and digital

Digital (87, 135, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 25, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (62, 110, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 55, 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]

(135−48, 135, 576)-Net in Base 8 — Constructive

(87, 135, 576)-net in base 8, using

1 times m-reduction [i] based on (87, 136, 576)-net in base 8, using
- trace code for nets [i] based on (19, 68, 288)-net in base 64, using
  - 2 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(135−48, 135, 1054)-Net over F₈ — Digital

Digital (87, 135, 1054)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−48, 135, 168309)-Net in Base 8 — Upper bound on s

There is no (87, 135, 168310)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 643708 486965 408655 995384 526575 042079 746412 392855 160971 849356 203600 078202 091291 231006 808059 459397 667970 217099 530641 077212 > 8¹³⁵ [i]