MinT - Best Known (4, 4+19, s)-Nets in Base 64

Best Known (4, 4+19, s)-Nets in Base 64

(4, 4+19, 104)-Net over F₆₄ — Constructive and digital

Digital (4, 23, 104)-net over F₆₄, using

t-expansion [i] based on digital (3, 23, 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]

(4, 4+19, 129)-Net in Base 64 — Constructive

(4, 23, 129)-net in base 64, using

5 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
- base change [i] based on digital (0, 24, 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]

(4, 4+19, 129)-Net over F₆₄ — Digital

Digital (4, 23, 129)-net over F₆₄, using

net from sequence [i] based on digital (4, 128)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 4 and N(F) ≥ 129, using
  - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(4, 4+19, 1707)-Net in Base 64 — Upper bound on s

There is no (4, 23, 1708)-net in base 64, because

1 times m-reduction [i] would yield (4, 22, 1708)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 5446 991278 998645 754880 788394 844740 302117 > 64²² [i]