MinT - Best Known (109−85, 109, s)-Nets in Base 16

Best Known (109−85, 109, s)-Nets in Base 16

(109−85, 109, 65)-Net over F₁₆ — Constructive and digital

Digital (24, 109, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 109, 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]

(109−85, 109, 129)-Net over F₁₆ — Digital

Digital (24, 109, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 109, 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]

(109−85, 109, 1350)-Net in Base 16 — Upper bound on s

There is no (24, 109, 1351)-net in base 16, because

1 times m-reduction [i] would yield (24, 108, 1351)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11124 619063 403872 378512 579221 968975 305289 429465 595603 238200 803440 570159 609786 269907 436116 555266 368529 682616 202190 160200 785882 754856 > 16¹⁰⁸ [i]