MinT - Best Known (82−63, 82, s)-Nets in Base 16

Best Known (82−63, 82, s)-Nets in Base 16

(82−63, 82, 65)-Net over F₁₆ — Constructive and digital

Digital (19, 82, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 82, 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]

(82−63, 82, 129)-Net over F₁₆ — Digital

Digital (19, 82, 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]

(82−63, 82, 1142)-Net in Base 16 — Upper bound on s

There is no (19, 82, 1143)-net in base 16, because

1 times m-reduction [i] would yield (19, 81, 1143)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 34 837798 627998 443673 703722 565310 365668 362550 979074 446997 746655 564427 190582 476353 553181 213144 550496 > 16⁸¹ [i]