MinT - Best Known (69−56, 69, s)-Nets in Base 64

Best Known (69−56, 69, s)-Nets in Base 64

(69−56, 69, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 69, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 69, 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]

(69−56, 69, 192)-Net in Base 64 — Constructive

(13, 69, 192)-net in base 64, using

1 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
- base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(69−56, 69, 257)-Net over F₆₄ — Digital

Digital (13, 69, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 69, 257)-net over F₆₄, using
- net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(69−56, 69, 5051)-Net in Base 64 — Upper bound on s

There is no (13, 69, 5052)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 42449 058192 148379 746863 921355 426579 922785 602387 821985 386887 304577 450744 338119 015706 702867 452901 380862 991015 451555 509543 711880 > 64⁶⁹ [i]