MinT - Best Known (63−43, 63, s)-Nets in Base 64

Best Known (63−43, 63, s)-Nets in Base 64

(63−43, 63, 177)-Net over F₆₄ — Constructive and digital

Digital (20, 63, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 63, 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]

(63−43, 63, 288)-Net in Base 64 — Constructive

(20, 63, 288)-net in base 64, using

14 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]

(63−43, 63, 342)-Net over F₆₄ — Digital

Digital (20, 63, 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]

(63−43, 63, 29616)-Net in Base 64 — Upper bound on s

There is no (20, 63, 29617)-net in base 64, because

1 times m-reduction [i] would yield (20, 62, 29617)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 9623 903044 690174 188660 143142 512770 935592 600679 106425 192148 891028 575423 057096 291505 873031 355383 017266 620008 390048 > 64⁶² [i]