MinT - Best Known (87−65, 87, s)-Nets in Base 64

Best Known (87−65, 87, s)-Nets in Base 64

(87−65, 87, 177)-Net over F₆₄ — Constructive and digital

Digital (22, 87, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 87, 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]

(87−65, 87, 288)-Net in Base 64 — Constructive

(22, 87, 288)-net in base 64, using

4 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]

(87−65, 87, 342)-Net over F₆₄ — Digital

Digital (22, 87, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 87, 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]

(87−65, 87, 14493)-Net in Base 64 — Upper bound on s

There is no (22, 87, 14494)-net in base 64, because

1 times m-reduction [i] would yield (22, 86, 14494)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 214826 223407 037691 248704 432380 871597 007618 398678 714804 233214 489148 101070 741158 005774 916580 207726 446419 983376 208394 179466 876585 174924 316822 144892 813254 474660 > 64⁸⁶ [i]