MinT - Best Known (69−49, 69, s)-Nets in Base 64

Best Known (69−49, 69, s)-Nets in Base 64

(69−49, 69, 177)-Net over F₆₄ — Constructive and digital

Digital (20, 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]

(69−49, 69, 288)-Net in Base 64 — Constructive

(20, 69, 288)-net in base 64, using

8 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]

(69−49, 69, 342)-Net over F₆₄ — Digital

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]

(69−49, 69, 20383)-Net in Base 64 — Upper bound on s

There is no (20, 69, 20384)-net in base 64, because

1 times m-reduction [i] would yield (20, 68, 20384)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 661 433285 691022 971574 434313 004301 765327 530218 093349 848014 386494 113300 223900 860050 112950 100018 974455 528679 123680 404160 642351 > 64⁶⁸ [i]