MinT - Best Known (27, 27+61, s)-Nets in Base 64

Best Known (27, 27+61, s)-Nets in Base 64

(27, 27+61, 177)-Net over F₆₄ — Constructive and digital

Digital (27, 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]

(27, 27+61, 288)-Net in Base 64 — Constructive

(27, 88, 288)-net in base 64, using

t-expansion [i] based on (22, 88, 288)-net in base 64, using
- 3 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]

(27, 27+61, 425)-Net over F₆₄ — Digital

Digital (27, 88, 425)-net over F₆₄, using

t-expansion [i] based on digital (26, 88, 425)-net over F₆₄, using
- net from sequence [i] based on digital (26, 424)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 26 and N(F) ≥ 425, using
    - a curve from the manYPoints database [i]

(27, 27+61, 33050)-Net in Base 64 — Upper bound on s

There is no (27, 88, 33051)-net in base 64, because

1 times m-reduction [i] would yield (27, 87, 33051)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 13 740303 937113 565473 240130 780423 966665 212437 165456 338046 387789 682597 980212 034993 810720 653380 146251 645651 767753 768807 045646 098060 852394 959983 849884 472978 836160 > 64⁸⁷ [i]