MinT - Best Known (108−31, 108, s)-Nets in Base 8

Best Known (108−31, 108, s)-Nets in Base 8

(108−31, 108, 514)-Net over F₈ — Constructive and digital

Digital (77, 108, 514)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (17, 32, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 16, 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 (45, 76, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 38, 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]

(108−31, 108, 576)-Net in Base 8 — Constructive

(77, 108, 576)-net in base 8, using

10 times m-reduction [i] based on (77, 118, 576)-net in base 8, using
- trace code for nets [i] based on (18, 59, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(108−31, 108, 3083)-Net over F₈ — Digital

Digital (77, 108, 3083)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(108−31, 108, 2539270)-Net in Base 8 — Upper bound on s

There is no (77, 108, 2539271)-net in base 8, because

1 times m-reduction [i] would yield (77, 107, 2539271)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 271990 090622 222390 714165 390547 530693 326182 760836 345543 935321 477277 117895 936555 495244 219224 285696 > 8¹⁰⁷ [i]