MinT - Best Known (86−67, 86, s)-Nets in Base 64

Best Known (86−67, 86, s)-Nets in Base 64

(86−67, 86, 177)-Net over F₆₄ — Constructive and digital

Digital (19, 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]

(86−67, 86, 216)-Net in Base 64 — Constructive

(19, 86, 216)-net in base 64, using

t-expansion [i] based on (18, 86, 216)-net in base 64, using
- 5 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]

(86−67, 86, 315)-Net over F₆₄ — Digital

Digital (19, 86, 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]

(86−67, 86, 9365)-Net in Base 64 — Upper bound on s

There is no (19, 86, 9366)-net in base 64, because

1 times m-reduction [i] would yield (19, 85, 9366)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3358 094866 389871 252124 735232 685555 855919 490385 897396 343168 290679 640088 210130 529914 107962 705876 291023 906069 537301 379709 950579 762419 477386 362097 210996 900092 > 64⁸⁵ [i]