MinT - Best Known (86−69, 86, s)-Nets in Base 64

Best Known (86−69, 86, s)-Nets in Base 64

(86−69, 86, 177)-Net over F₆₄ — Constructive and digital

Digital (17, 86, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 86, 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]

(86−69, 86, 192)-Net in Base 64 — Constructive

(17, 86, 192)-net in base 64, using

t-expansion [i] based on (16, 86, 192)-net in base 64, using
- 5 times m-reduction [i] based on (16, 91, 192)-net in base 64, using
  - base change [i] based on digital (3, 78, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(86−69, 86, 267)-Net over F₆₄ — Digital

Digital (17, 86, 267)-net over F₆₄, using

t-expansion [i] based on digital (16, 86, 267)-net over F₆₄, using
- net from sequence [i] based on digital (16, 266)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 16 and N(F) ≥ 267, using
    - a curve from the manYPoints database [i]

(86−69, 86, 7023)-Net in Base 64 — Upper bound on s

There is no (17, 86, 7024)-net in base 64, because

1 times m-reduction [i] would yield (17, 85, 7024)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3365 487487 179364 331550 621358 321844 796074 262223 668322 523902 607289 968578 886168 268048 696660 017617 213728 069282 123453 539862 849796 121623 182276 602901 475248 746011 > 64⁸⁵ [i]