MinT - Best Known (55−35, 55, s)-Nets in Base 16

Best Known (55−35, 55, s)-Nets in Base 16

(55−35, 55, 65)-Net over F₁₆ — Constructive and digital

Digital (20, 55, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 55, 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]

(55−35, 55, 104)-Net in Base 16 — Constructive

(20, 55, 104)-net in base 16, using

base change [i] based on digital (9, 44, 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]

(55−35, 55, 129)-Net over F₁₆ — Digital

Digital (20, 55, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 55, 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]

(55−35, 55, 3187)-Net in Base 16 — Upper bound on s

There is no (20, 55, 3188)-net in base 16, because

1 times m-reduction [i] would yield (20, 54, 3188)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 105535 840286 011635 477538 957901 612537 426951 804725 602442 005347 108791 > 16⁵⁴ [i]