MinT - Best Known (67−43, 67, s)-Nets in Base 64

Best Known (67−43, 67, s)-Nets in Base 64

(67−43, 67, 177)-Net over F₆₄ — Constructive and digital

Digital (24, 67, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 67, 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]

(67−43, 67, 288)-Net in Base 64 — Constructive

(24, 67, 288)-net in base 64, using

t-expansion [i] based on (22, 67, 288)-net in base 64, using
- 24 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]

(67−43, 67, 342)-Net over F₆₄ — Digital

Digital (24, 67, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 67, 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]

(67−43, 67, 353)-Net in Base 64

(24, 67, 353)-net in base 64, using

3 times m-reduction [i] based on (24, 70, 353)-net in base 64, using
- base change [i] based on digital (14, 60, 353)-net over F₁₂₈, using
  - net from sequence [i] based on digital (14, 352)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 14 and N(F) ≥ 353, using
      - a curve from the manYPoints database [i]

(67−43, 67, 65410)-Net in Base 64 — Upper bound on s

There is no (24, 67, 65411)-net in base 64, because

1 times m-reduction [i] would yield (24, 66, 65411)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 161404 186009 352393 399805 300120 840311 527212 432309 635199 877442 740054 429224 785177 886293 115795 737438 273007 918540 651683 164864 > 64⁶⁶ [i]