MinT - Best Known (89−73, 89, s)-Nets in Base 16

Best Known (89−73, 89, s)-Nets in Base 16

(89−73, 89, 65)-Net over F₁₆ — Constructive and digital

Digital (16, 89, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 89, 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]

(89−73, 89, 98)-Net over F₁₆ — Digital

Digital (16, 89, 98)-net over F₁₆, using

t-expansion [i] based on digital (15, 89, 98)-net over F₁₆, using
- net from sequence [i] based on digital (15, 97)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 15 and N(F) ≥ 98, using
    - a curve from the manYPoints database [i]

(89−73, 89, 815)-Net in Base 16 — Upper bound on s

There is no (16, 89, 816)-net in base 16, because

1 times m-reduction [i] would yield (16, 88, 816)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9206 842574 859837 487621 211596 454496 873553 303729 001221 967556 473130 973423 610804 597783 595753 371143 420652 003641 > 16⁸⁸ [i]