MinT - Best Known (55−31, 55, s)-Nets in Base 64

Best Known (55−31, 55, s)-Nets in Base 64

(55−31, 55, 257)-Net over F₆₄ — Constructive and digital

Digital (24, 55, 257)-net over F₆₄, using

1 times m-reduction [i] based on digital (24, 56, 257)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 17, 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 (7, 39, 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]

(55−31, 55, 288)-Net in Base 64 — Constructive

(24, 55, 288)-net in base 64, using

t-expansion [i] based on (22, 55, 288)-net in base 64, using
- 36 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]

(55−31, 55, 407)-Net over F₆₄ — Digital

Digital (24, 55, 407)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(55−31, 55, 513)-Net in Base 64

(24, 55, 513)-net in base 64, using

9 times m-reduction [i] based on (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]

(55−31, 55, 324089)-Net in Base 64 — Upper bound on s

There is no (24, 55, 324090)-net in base 64, because

1 times m-reduction [i] would yield (24, 54, 324090)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 34 177365 344211 156067 132256 248126 082908 881083 846622 131234 113826 669284 517834 019518 290550 751158 446840 > 64⁵⁴ [i]