MinT - Best Known (90−69, 90, s)-Nets in Base 16

Best Known (90−69, 90, s)-Nets in Base 16

(90−69, 90, 65)-Net over F₁₆ — Constructive and digital

Digital (21, 90, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 90, 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]

(90−69, 90, 129)-Net over F₁₆ — Digital

Digital (21, 90, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 90, 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]

(90−69, 90, 1261)-Net in Base 16 — Upper bound on s

There is no (21, 90, 1262)-net in base 16, because

1 times m-reduction [i] would yield (21, 89, 1262)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 147955 025668 126651 002723 331446 982089 902211 175736 348335 204569 376915 939407 422972 506083 891100 862565 268141 792546 > 16⁸⁹ [i]