MinT - Best Known (89−65, 89, s)-Nets in Base 64

Best Known (89−65, 89, s)-Nets in Base 64

(89−65, 89, 177)-Net over F₆₄ — Constructive and digital

Digital (24, 89, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 89, 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]

(89−65, 89, 288)-Net in Base 64 — Constructive

(24, 89, 288)-net in base 64, using

t-expansion [i] based on (22, 89, 288)-net in base 64, using
- 2 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
  - base change [i] based on digital (9, 78, 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]

(89−65, 89, 342)-Net over F₆₄ — Digital

Digital (24, 89, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 89, 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]

(89−65, 89, 18800)-Net in Base 64 — Upper bound on s

There is no (24, 89, 18801)-net in base 64, because

1 times m-reduction [i] would yield (24, 88, 18801)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 879 715061 114013 818784 711122 776711 028229 085474 535499 828217 695849 050248 556618 791203 841394 728725 426597 739830 599889 885350 314184 410907 708702 301383 295782 785141 224145 > 64⁸⁸ [i]