MinT - Best Known (127−43, 127, s)-Nets in Base 8

Best Known (127−43, 127, s)-Nets in Base 8

(127−43, 127, 382)-Net over F₈ — Constructive and digital

Digital (84, 127, 382)-net over F₈, using

8¹ times duplication [i] based on digital (83, 126, 382)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 26, 28)-net over F₈, using
    - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
        a function field by SÃ©mirat [i]
  2. digital (57, 100, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 50, 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]

(127−43, 127, 576)-Net in Base 8 — Constructive

(84, 127, 576)-net in base 8, using

3 times m-reduction [i] based on (84, 130, 576)-net in base 8, using
- trace code for nets [i] based on (19, 65, 288)-net in base 64, using
  - 5 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]

(127−43, 127, 1290)-Net over F₈ — Digital

Digital (84, 127, 1290)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−43, 127, 325025)-Net in Base 8 — Upper bound on s

There is no (84, 127, 325026)-net in base 8, because

1 times m-reduction [i] would yield (84, 126, 325026)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 615686 826211 858666 710765 669561 612683 629242 161873 549582 716738 878830 901162 317391 014105 095031 860824 010369 173953 614648 > 8¹²⁶ [i]