MinT - Best Known (18, 18+89, s)-Nets in Base 16

Best Known (18, 18+89, s)-Nets in Base 16

(18, 18+89, 65)-Net over F₁₆ — Constructive and digital

Digital (18, 107, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 107, 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]

(18, 18+89, 113)-Net over F₁₆ — Digital

Digital (18, 107, 113)-net over F₁₆, using

net from sequence [i] based on digital (18, 112)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 18 and N(F) ≥ 113, using
  - a curve from the manYPoints database [i]

(18, 18+89, 891)-Net in Base 16 — Upper bound on s

There is no (18, 107, 892)-net in base 16, because

1 times m-reduction [i] would yield (18, 106, 892)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 44 927744 760918 238889 201157 580521 458906 581196 190055 970239 696897 820913 723816 650562 734627 348566 423724 847267 313061 426954 142819 079396 > 16¹⁰⁶ [i]