MinT - Best Known (90, 127, s)-Nets in Base 8

Best Known (90, 127, s)-Nets in Base 8

(90, 127, 514)-Net over F₈ — Constructive and digital

Digital (90, 127, 514)-net over F₈, using

8¹ times duplication [i] based on digital (89, 126, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (20, 38, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 19, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. digital (51, 88, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 44, 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]

(90, 127, 590)-Net in Base 8 — Constructive

(90, 127, 590)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 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. (71, 108, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 54, 288)-net in base 64, using
    - 2 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
      - base change [i] based on digital (9, 48, 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]

(90, 127, 3148)-Net over F₈ — Digital

Digital (90, 127, 3148)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(90, 127, 2262871)-Net in Base 8 — Upper bound on s

There is no (90, 127, 2262872)-net in base 8, because

1 times m-reduction [i] would yield (90, 126, 2262872)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 615661 141138 519018 361925 961638 381775 006753 943990 386851 773915 793867 404599 618204 379398 141139 637757 767091 530184 898701 > 8¹²⁶ [i]