MinT - Best Known (69−54, 69, s)-Nets in Base 64

Best Known (69−54, 69, s)-Nets in Base 64

(69−54, 69, 177)-Net over F₆₄ — Constructive and digital

Digital (15, 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]

(69−54, 69, 216)-Net in Base 64 — Constructive

(15, 69, 216)-net in base 64, using

1 times m-reduction [i] based on (15, 70, 216)-net in base 64, using
- base change [i] based on digital (5, 60, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(69−54, 69, 258)-Net over F₆₄ — Digital

Digital (15, 69, 258)-net over F₆₄, using

net from sequence [i] based on digital (15, 257)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 15 and N(F) ≥ 258, using
  - a curve from the manYPoints database [i]

(69−54, 69, 7145)-Net in Base 64 — Upper bound on s

There is no (15, 69, 7146)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 42356 449320 320923 564590 972879 850070 459802 527712 843458 184784 866045 066617 199168 545338 243920 206399 965549 720497 962531 207051 696464 > 64⁶⁹ [i]