MinT - Best Known (62−39, 62, s)-Nets in Base 64

Best Known (62−39, 62, s)-Nets in Base 64

(62−39, 62, 184)-Net over F₆₄ — Constructive and digital

Digital (23, 62, 184)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 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 (3, 42, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(62−39, 62, 288)-Net in Base 64 — Constructive

(23, 62, 288)-net in base 64, using

t-expansion [i] based on (22, 62, 288)-net in base 64, using
- 29 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]

(62−39, 62, 342)-Net over F₆₄ — Digital

Digital (23, 62, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 62, 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]

(62−39, 62, 353)-Net in Base 64

(23, 62, 353)-net in base 64, using

1 times m-reduction [i] based on (23, 63, 353)-net in base 64, using
- base change [i] based on digital (14, 54, 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]

(62−39, 62, 79179)-Net in Base 64 — Upper bound on s

There is no (23, 62, 79180)-net in base 64, because

1 times m-reduction [i] would yield (23, 61, 79180)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 150 327881 253172 255356 296567 266802 261425 326410 053828 283898 055468 932142 265896 180785 212571 341623 450490 514398 303884 > 64⁶¹ [i]