MinT - Best Known (107−31, 107, s)-Nets in Base 8

Best Known (107−31, 107, s)-Nets in Base 8

(107−31, 107, 484)-Net over F₈ — Constructive and digital

Digital (76, 107, 484)-net over F₈, using

8¹ times duplication [i] based on digital (75, 106, 484)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 30, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 15, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [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]

(107−31, 107, 576)-Net in Base 8 — Constructive

(76, 107, 576)-net in base 8, using

9 times m-reduction [i] based on (76, 116, 576)-net in base 8, using
- trace code for nets [i] based on (18, 58, 288)-net in base 64, using
  - 5 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]

(107−31, 107, 2877)-Net over F₈ — Digital

Digital (76, 107, 2877)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(107−31, 107, 2210562)-Net in Base 8 — Upper bound on s

There is no (76, 107, 2210563)-net in base 8, because

1 times m-reduction [i] would yield (76, 106, 2210563)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 534000 153195 823572 145527 809131 911012 806082 614575 164369 884060 006368 383124 330981 576792 506932 498176 > 8¹⁰⁶ [i]