MinT - Best Known (25, 56, s)-Nets in Base 64

Best Known (25, 56, s)-Nets in Base 64

(25, 56, 281)-Net over F₆₄ — Constructive and digital

Digital (25, 56, 281)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 18, 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]
2. digital (7, 38, 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]

(25, 56, 306)-Net in Base 64 — Constructive

(25, 56, 306)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (3, 18, 129)-net in base 64, using
  - 3 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
    - base change [i] based on digital (0, 18, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
2. digital (7, 38, 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]

(25, 56, 465)-Net over F₆₄ — Digital

Digital (25, 56, 465)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(25, 56, 513)-Net in Base 64

(25, 56, 513)-net in base 64, using

12 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]

(25, 56, 427640)-Net in Base 64 — Upper bound on s

There is no (25, 56, 427641)-net in base 64, because

1 times m-reduction [i] would yield (25, 55, 427641)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2187 302939 054620 865985 308294 709943 167901 545487 028142 679882 583485 118922 760657 670888 340955 521943 506352 > 64⁵⁵ [i]