MinT - Best Known (20, 20+51, s)-Nets in Base 64

Best Known (20, 20+51, s)-Nets in Base 64

(20, 20+51, 177)-Net over F₆₄ — Constructive and digital

Digital (20, 71, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 71, 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+51, 288)-Net in Base 64 — Constructive

(20, 71, 288)-net in base 64, using

6 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+51, 342)-Net over F₆₄ — Digital

Digital (20, 71, 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+51, 18420)-Net in Base 64 — Upper bound on s

There is no (20, 71, 18421)-net in base 64, because

1 times m-reduction [i] would yield (20, 70, 18421)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2 710200 268970 822789 583898 088957 912161 947775 581694 251224 167301 469862 555237 255891 938006 610006 477014 670008 382609 165637 532023 292592 > 64⁷⁰ [i]