MinT - Best Known (20, 49, s)-Nets in Base 64

Best Known (20, 49, s)-Nets in Base 64

(20, 49, 208)-Net over F₆₄ — Constructive and digital

Digital (20, 49, 208)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 17, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]
2. digital (3, 32, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄ (see above)

(20, 49, 288)-Net in Base 64 — Constructive

(20, 49, 288)-net in base 64, using

28 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
- base change [i] based on digital (9, 66, 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]

(20, 49, 342)-Net over F₆₄ — Digital

Digital (20, 49, 342)-net over F₆₄, using

net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
  - a curve from the manYPoints database [i]

(20, 49, 149525)-Net in Base 64 — Upper bound on s

There is no (20, 49, 149526)-net in base 64, because

1 times m-reduction [i] would yield (20, 48, 149526)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 497 355234 180815 443143 387287 898835 473118 247489 517953 882484 321375 132104 614419 635794 353976 > 64⁴⁸ [i]