MinT - Best Known (51−38, 51, s)-Nets in Base 64

Best Known (51−38, 51, s)-Nets in Base 64

(51−38, 51, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 51, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 51, 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]

(51−38, 51, 257)-Net in Base 64 — Constructive

(13, 51, 257)-net in base 64, using

1 times m-reduction [i] based on (13, 52, 257)-net in base 64, using
- base change [i] based on digital (0, 39, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(51−38, 51, 257)-Net over F₆₄ — Digital

Digital (13, 51, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 51, 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]

(51−38, 51, 8863)-Net in Base 64 — Upper bound on s

There is no (13, 51, 8864)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 130 615154 315932 299070 344621 015958 127733 856177 923773 651792 876177 091474 518749 883192 047701 306235 > 64⁵¹ [i]