MinT - Best Known (19, 48, s)-Nets in Base 64

Best Known (19, 48, s)-Nets in Base 64

(19, 48, 193)-Net over F₆₄ — Constructive and digital

Digital (19, 48, 193)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
2. digital (5, 34, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(19, 48, 288)-Net in Base 64 — Constructive

(19, 48, 288)-net in base 64, using

22 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
- base change [i] based on digital (9, 60, 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]

(19, 48, 315)-Net over F₆₄ — Digital

Digital (19, 48, 315)-net over F₆₄, using

net from sequence [i] based on digital (19, 314)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 19 and N(F) ≥ 315, using
  - a curve from the manYPoints database [i]

(19, 48, 321)-Net in Base 64

(19, 48, 321)-net in base 64, using

1 times m-reduction [i] based on (19, 49, 321)-net in base 64, using
- base change [i] based on digital (12, 42, 321)-net over F₁₂₈, using
  - net from sequence [i] based on digital (12, 320)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 12 and N(F) ≥ 321, using
      - a curve from the manYPoints database [i]

(19, 48, 111095)-Net in Base 64 — Upper bound on s

There is no (19, 48, 111096)-net in base 64, because

1 times m-reduction [i] would yield (19, 47, 111096)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 7 771505 108300 330421 132923 173190 066902 062186 620661 265211 886726 818233 912758 372466 765786 > 64⁴⁷ [i]