MinT - Best Known (88−69, 88, s)-Nets in Base 64

Best Known (88−69, 88, s)-Nets in Base 64

(88−69, 88, 177)-Net over F₆₄ — Constructive and digital

Digital (19, 88, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 88, 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]

(88−69, 88, 216)-Net in Base 64 — Constructive

(19, 88, 216)-net in base 64, using

t-expansion [i] based on (18, 88, 216)-net in base 64, using
- 3 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
  - base change [i] based on digital (5, 78, 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]

(88−69, 88, 315)-Net over F₆₄ — Digital

Digital (19, 88, 315)-net over F₆₄, using

net from sequence [i] based on digital (19, 314)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 19 and N(F) ≥ 315, using
  - a curve from the manYPoints database [i]

(88−69, 88, 8974)-Net in Base 64 — Upper bound on s

There is no (19, 88, 8975)-net in base 64, because

1 times m-reduction [i] would yield (19, 87, 8975)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 13 758875 541579 512229 123686 255232 419128 644002 696829 948846 928485 996764 543396 529993 589288 685967 400683 750865 770426 567671 096332 535771 735969 453732 283679 563997 874008 > 64⁸⁷ [i]