MinT - Best Known (66−45, 66, s)-Nets in Base 64

Best Known (66−45, 66, s)-Nets in Base 64

(66−45, 66, 177)-Net over F₆₄ — Constructive and digital

Digital (21, 66, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 66, 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]

(66−45, 66, 288)-Net in Base 64 — Constructive

(21, 66, 288)-net in base 64, using

18 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
- base change [i] based on digital (9, 72, 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]

(66−45, 66, 342)-Net over F₆₄ — Digital

Digital (21, 66, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 66, 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]

(66−45, 66, 31174)-Net in Base 64 — Upper bound on s

There is no (21, 66, 31175)-net in base 64, because

1 times m-reduction [i] would yield (21, 65, 31175)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2522 271905 898122 279910 116298 003762 289743 613057 889256 576340 357166 499935 887270 021645 625700 376749 791878 479251 486950 875316 > 64⁶⁵ [i]