MinT - Best Known (25, 25+85, s)-Nets in Base 16

Best Known (25, 25+85, s)-Nets in Base 16

(25, 25+85, 65)-Net over F₁₆ — Constructive and digital

Digital (25, 110, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 110, 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]

(25, 25+85, 66)-Net in Base 16 — Constructive

(25, 110, 66)-net in base 16, using

net from sequence [i] based on (25, 65)-sequence in base 16, using
- base expansion [i] based on digital (50, 65)-sequence over F₄, using
  - t-expansion [i] based on digital (49, 65)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
      - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(25, 25+85, 144)-Net over F₁₆ — Digital

Digital (25, 110, 144)-net over F₁₆, using

net from sequence [i] based on digital (25, 143)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 25 and N(F) ≥ 144, using
  - a curve from the manYPoints database [i]

(25, 25+85, 1444)-Net in Base 16 — Upper bound on s

There is no (25, 110, 1445)-net in base 16, because

1 times m-reduction [i] would yield (25, 109, 1445)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 179160 325543 190632 227496 449155 382552 292668 511193 133804 982660 071401 127639 732035 733321 420789 319483 122991 880110 479490 311008 146348 266976 > 16¹⁰⁹ [i]