MinT - Best Known (69−48, 69, s)-Nets in Base 64

Best Known (69−48, 69, s)-Nets in Base 64

(69−48, 69, 177)-Net over F₆₄ — Constructive and digital

Digital (21, 69, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 69, 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]

(69−48, 69, 288)-Net in Base 64 — Constructive

(21, 69, 288)-net in base 64, using

15 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]

(69−48, 69, 342)-Net over F₆₄ — Digital

Digital (21, 69, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 69, 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]

(69−48, 69, 24242)-Net in Base 64 — Upper bound on s

There is no (21, 69, 24243)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 42329 832524 431671 193817 394762 040965 286810 217056 736046 159981 392185 020767 905575 795873 082347 100349 525369 868503 534408 126218 148764 > 64⁶⁹ [i]