MinT - Best Known (61−35, 61, s)-Nets in Base 64

Best Known (61−35, 61, s)-Nets in Base 64

(61−35, 61, 257)-Net over F₆₄ — Constructive and digital

Digital (26, 61, 257)-net over F₆₄, using

1 times m-reduction [i] based on digital (26, 62, 257)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 19, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. digital (7, 43, 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]

(61−35, 61, 288)-Net in Base 64 — Constructive

(26, 61, 288)-net in base 64, using

t-expansion [i] based on (22, 61, 288)-net in base 64, using
- 30 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]

(61−35, 61, 425)-Net over F₆₄ — Digital

Digital (26, 61, 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]

(61−35, 61, 513)-Net in Base 64

(26, 61, 513)-net in base 64, using

11 times m-reduction [i] based on (26, 72, 513)-net in base 64, using
- base change [i] based on digital (8, 54, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(61−35, 61, 269987)-Net in Base 64 — Upper bound on s

There is no (26, 61, 269988)-net in base 64, because

1 times m-reduction [i] would yield (26, 60, 269988)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2 348566 246407 401818 188567 654984 725669 680337 302983 298839 037544 960578 638371 862732 632451 739232 192201 648538 457750 > 64⁶⁰ [i]