MinT - Best Known (61−39, 61, s)-Nets in Base 16

Best Known (61−39, 61, s)-Nets in Base 16

(61−39, 61, 65)-Net over F₁₆ — Constructive and digital

Digital (22, 61, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 61, 65)-net over F₁₆, using
- net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
    - the Hermitian function field over F₁₆ [i]

(61−39, 61, 104)-Net in Base 16 — Constructive

(22, 61, 104)-net in base 16, using

4 times m-reduction [i] based on (22, 65, 104)-net in base 16, using
- base change [i] based on digital (9, 52, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(61−39, 61, 129)-Net over F₁₆ — Digital

Digital (22, 61, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 61, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(61−39, 61, 3344)-Net in Base 16 — Upper bound on s

There is no (22, 61, 3345)-net in base 16, because

1 times m-reduction [i] would yield (22, 60, 3345)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 775960 141206 104485 616761 598501 084903 326655 037406 073717 836962 428289 720576 > 16⁶⁰ [i]