MinT - Best Known (111−99, 111, s)-Nets in Base 16

Best Known (111−99, 111, s)-Nets in Base 16

(111−99, 111, 65)-Net over F₁₆ — Constructive and digital

Digital (12, 111, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 111, 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]

(111−99, 111, 88)-Net over F₁₆ — Digital

Digital (12, 111, 88)-net over F₁₆, using

net from sequence [i] based on digital (12, 87)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 12 and N(F) ≥ 88, using
  - a curve from the manYPoints database [i]

(111−99, 111, 594)-Net in Base 16 — Upper bound on s

There is no (12, 111, 595)-net in base 16, because

23 times m-reduction [i] would yield (12, 88, 595)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9400 768965 840589 566962 455820 698623 788623 369809 843677 277692 740557 179063 436439 213356 387766 799494 253628 242776 > 16⁸⁸ [i]