MinT - Best Known (126−111, 126, s)-Nets in Base 16

Best Known (126−111, 126, s)-Nets in Base 16

(126−111, 126, 65)-Net over F₁₆ — Constructive and digital

Digital (15, 126, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 126, 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]

(126−111, 126, 98)-Net over F₁₆ — Digital

Digital (15, 126, 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]

(126−111, 126, 733)-Net in Base 16 — Upper bound on s

There is no (15, 126, 734)-net in base 16, because

17 times m-reduction [i] would yield (15, 109, 734)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 181774 059527 504063 496153 399885 158151 294927 929792 503907 266447 278526 887814 641776 917030 813132 336301 434360 858497 416427 594931 424256 970096 > 16¹⁰⁹ [i]