MinT - Best Known (57−45, 57, s)-Nets in Base 64

Best Known (57−45, 57, s)-Nets in Base 64

(57−45, 57, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 57, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 57, 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]

(57−45, 57, 192)-Net in Base 64 — Constructive

(12, 57, 192)-net in base 64, using

6 times m-reduction [i] based on (12, 63, 192)-net in base 64, using
- base change [i] based on digital (3, 54, 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]

(57−45, 57, 257)-Net over F₆₄ — Digital

Digital (12, 57, 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]

(57−45, 57, 5678)-Net in Base 64 — Upper bound on s

There is no (12, 57, 5679)-net in base 64, because

1 times m-reduction [i] would yield (12, 56, 5679)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 140260 411046 695472 764526 504378 854252 347050 566696 695157 001166 675276 474732 899340 822732 955371 755301 948670 > 64⁵⁶ [i]