MinT - Best Known (54−29, 54, s)-Nets in Base 128

Best Known (54−29, 54, s)-Nets in Base 128

(54−29, 54, 438)-Net over F₁₂₈ — Constructive and digital

Digital (25, 54, 438)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (25, 55, 438)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 16, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. digital (9, 39, 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]

(54−29, 54, 545)-Net in Base 128 — Constructive

(25, 54, 545)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (2, 16, 257)-net in base 128, using
  - base change [i] based on digital (0, 14, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]
2. digital (9, 38, 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]

(54−29, 54, 1045)-Net over F₁₂₈ — Digital

Digital (25, 54, 1045)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−29, 54, 4518021)-Net in Base 128 — Upper bound on s

There is no (25, 54, 4518022)-net in base 128, because

1 times m-reduction [i] would yield (25, 53, 4518022)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 4809 826554 926772 406520 818867 311824 508104 968792 844852 129983 958506 132237 220455 506672 319152 382505 940618 294343 559232 > 128⁵³ [i]