MinT - Best Known (60−37, 60, s)-Nets in Base 64

Best Known (60−37, 60, s)-Nets in Base 64

(60−37, 60, 193)-Net over F₆₄ — Constructive and digital

Digital (23, 60, 193)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 18, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
2. digital (5, 42, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(60−37, 60, 288)-Net in Base 64 — Constructive

(23, 60, 288)-net in base 64, using

t-expansion [i] based on (22, 60, 288)-net in base 64, using
- 31 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]

(60−37, 60, 342)-Net over F₆₄ — Digital

Digital (23, 60, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 60, 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]

(60−37, 60, 513)-Net in Base 64

(23, 60, 513)-net in base 64, using

base change [i] based on digital (8, 45, 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]

(60−37, 60, 99771)-Net in Base 64 — Upper bound on s

There is no (23, 60, 99772)-net in base 64, because

1 times m-reduction [i] would yield (23, 59, 99772)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 36697 240414 237334 081765 733912 904453 675955 129635 129490 192598 702068 968616 279944 092443 230678 678789 220651 324503 > 64⁵⁹ [i]