MinT - Best Known (13, 40, s)-Nets in Base 16

Best Known (13, 40, s)-Nets in Base 16

(13, 40, 65)-Net over F₁₆ — Constructive and digital

Digital (13, 40, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 40, 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]

(13, 40, 76)-Net in Base 16 — Constructive

(13, 40, 76)-net in base 16, using

base change [i] based on digital (5, 32, 76)-net over F₃₂, using
- net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
    - a function field by SÃ©mirat [i]

(13, 40, 97)-Net over F₁₆ — Digital

Digital (13, 40, 97)-net over F₁₆, using

net from sequence [i] based on digital (13, 96)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 13 and N(F) ≥ 97, using
  - a curve from the manYPoints database [i]

(13, 40, 1540)-Net in Base 16 — Upper bound on s

There is no (13, 40, 1541)-net in base 16, because

1 times m-reduction [i] would yield (13, 39, 1541)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 91498 219230 181997 117749 990966 853661 105422 711296 > 16³⁹ [i]