MinT - Best Known (52−32, 52, s)-Nets in Base 64

Best Known (52−32, 52, s)-Nets in Base 64

(52−32, 52, 184)-Net over F₆₄ — Constructive and digital

Digital (20, 52, 184)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]
2. digital (3, 35, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(52−32, 52, 288)-Net in Base 64 — Constructive

(20, 52, 288)-net in base 64, using

25 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]

(52−32, 52, 342)-Net over F₆₄ — Digital

Digital (20, 52, 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]

(52−32, 52, 80027)-Net in Base 64 — Upper bound on s

There is no (20, 52, 80028)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 8343 991099 941457 321755 239055 485362 410426 271764 319248 180408 273875 821537 204577 036065 582983 933775 > 64⁵² [i]