MinT - Best Known (24, 64, s)-Nets in Base 64

Best Known (24, 64, s)-Nets in Base 64

(24, 64, 184)-Net over F₆₄ — Constructive and digital

Digital (24, 64, 184)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 21, 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, 43, 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]

(24, 64, 288)-Net in Base 64 — Constructive

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

t-expansion [i] based on (22, 64, 288)-net in base 64, using
- 27 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]

(24, 64, 342)-Net over F₆₄ — Digital

Digital (24, 64, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 64, 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]

(24, 64, 513)-Net in Base 64

(24, 64, 513)-net in base 64, using

base change [i] based on digital (8, 48, 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]

(24, 64, 79375)-Net in Base 64 — Upper bound on s

There is no (24, 64, 79376)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 39 404404 553452 141021 696936 814379 008324 069143 471494 530639 758372 174566 999451 884523 441805 662732 678202 412118 408000 228576 > 64⁶⁴ [i]