MinT - Best Known (87−63, 87, s)-Nets in Base 64

Best Known (87−63, 87, s)-Nets in Base 64

(87−63, 87, 177)-Net over F₆₄ — Constructive and digital

Digital (24, 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]

(87−63, 87, 288)-Net in Base 64 — Constructive

(24, 87, 288)-net in base 64, using

t-expansion [i] based on (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]

(87−63, 87, 342)-Net over F₆₄ — Digital

Digital (24, 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]

(87−63, 87, 20186)-Net in Base 64 — Upper bound on s

There is no (24, 87, 20187)-net in base 64, because

1 times m-reduction [i] would yield (24, 86, 20187)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 214826 071804 178474 692611 645845 592227 925842 765193 660975 534183 245564 180083 977189 244588 152027 906162 695294 645454 735137 298954 244459 796549 418426 491747 357180 666560 > 64⁸⁶ [i]