MinT - Best Known (90−69, 90, s)-Nets in Base 64

Best Known (90−69, 90, s)-Nets in Base 64

(90−69, 90, 177)-Net over F₆₄ — Constructive and digital

Digital (21, 90, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 90, 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]

(90−69, 90, 216)-Net in Base 64 — Constructive

(21, 90, 216)-net in base 64, using

t-expansion [i] based on (18, 90, 216)-net in base 64, using
- 1 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]

(90−69, 90, 342)-Net over F₆₄ — Digital

Digital (21, 90, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 90, 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]

(90−69, 90, 11466)-Net in Base 64 — Upper bound on s

There is no (21, 90, 11467)-net in base 64, because

1 times m-reduction [i] would yield (21, 89, 11467)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 56315 371150 045593 542855 507003 442206 024367 520464 116636 921337 351187 328986 627635 257874 299243 653143 791255 868751 371455 565633 628711 758061 812565 424011 371727 778906 683186 > 64⁸⁹ [i]