MinT - Best Known (10, 10+31, s)-Nets in Base 16

Best Known (10, 10+31, s)-Nets in Base 16

(10, 10+31, 65)-Net over F₁₆ — Constructive and digital

Digital (10, 41, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 41, 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]

(10, 10+31, 81)-Net over F₁₆ — Digital

Digital (10, 41, 81)-net over F₁₆, using

net from sequence [i] based on digital (10, 80)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 10 and N(F) ≥ 81, using
  - a curve from the manYPoints database [i]

(10, 10+31, 688)-Net in Base 16 — Upper bound on s

There is no (10, 41, 689)-net in base 16, because

1 times m-reduction [i] would yield (10, 40, 689)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 489828 764432 593773 793028 216027 730384 603949 813776 > 16⁴⁰ [i]