MinT - Best Known (67−42, 67, s)-Nets in Base 64

Best Known (67−42, 67, s)-Nets in Base 64

(67−42, 67, 184)-Net over F₆₄ — Constructive and digital

Digital (25, 67, 184)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 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, 45, 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]

(67−42, 67, 288)-Net in Base 64 — Constructive

(25, 67, 288)-net in base 64, using

t-expansion [i] based on (22, 67, 288)-net in base 64, using
- 24 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]

(67−42, 67, 408)-Net over F₆₄ — Digital

Digital (25, 67, 408)-net over F₆₄, using

net from sequence [i] based on digital (25, 407)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 25 and N(F) ≥ 408, using
  - a curve from the manYPoints database [i]

(67−42, 67, 513)-Net in Base 64

(25, 67, 513)-net in base 64, using

1 times m-reduction [i] based on (25, 68, 513)-net in base 64, using
- base change [i] based on digital (8, 51, 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]

(67−42, 67, 79738)-Net in Base 64 — Upper bound on s

There is no (25, 67, 79739)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 10 329418 704685 203373 106831 595795 860993 544554 116264 232025 494370 515512 875339 371706 284275 087792 054564 030787 294546 176682 463508 > 64⁶⁷ [i]