MinT - Best Known (20, 20+53, s)-Nets in Base 64

Best Known (20, 20+53, s)-Nets in Base 64

(20, 20+53, 177)-Net over F₆₄ — Constructive and digital

Digital (20, 73, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 73, 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]

(20, 20+53, 288)-Net in Base 64 — Constructive

(20, 73, 288)-net in base 64, using

4 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, 20+53, 342)-Net over F₆₄ — Digital

Digital (20, 73, 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, 20+53, 16801)-Net in Base 64 — Upper bound on s

There is no (20, 73, 16802)-net in base 64, because

1 times m-reduction [i] would yield (20, 72, 16802)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 11099 258453 486737 300746 881314 447646 939170 231967 930488 409590 379508 547847 797918 942903 735635 875979 498994 057396 177450 128755 475302 665840 > 64⁷² [i]