MinT - Best Known (68−45, 68, s)-Nets in Base 64

Best Known (68−45, 68, s)-Nets in Base 64

(68−45, 68, 177)-Net over F₆₄ — Constructive and digital

Digital (23, 68, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 68, 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]

(68−45, 68, 288)-Net in Base 64 — Constructive

(23, 68, 288)-net in base 64, using

t-expansion [i] based on (22, 68, 288)-net in base 64, using
- 23 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
  - base change [i] based on digital (9, 78, 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]

(68−45, 68, 342)-Net over F₆₄ — Digital

Digital (23, 68, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 68, 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]

(68−45, 68, 45503)-Net in Base 64 — Upper bound on s

There is no (23, 68, 45504)-net in base 64, because

1 times m-reduction [i] would yield (23, 67, 45504)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 10 329844 084756 973957 138198 547555 327561 457427 007822 743512 085557 597974 208498 857453 780976 902812 633069 123404 540792 932958 423645 > 64⁶⁷ [i]