MinT - Best Known (109−82, 109, s)-Nets in Base 16

Best Known (109−82, 109, s)-Nets in Base 16

(109−82, 109, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 109, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 109, 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]

(109−82, 109, 76)-Net in Base 16 — Constructive

(27, 109, 76)-net in base 16, using

1 times m-reduction [i] based on (27, 110, 76)-net in base 16, using
- base change [i] based on digital (5, 88, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(109−82, 109, 156)-Net over F₁₆ — Digital

Digital (27, 109, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(109−82, 109, 1687)-Net in Base 16 — Upper bound on s

There is no (27, 109, 1688)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 179891 240737 233757 010130 336020 388378 363972 173972 136082 968763 368790 468896 305121 981343 391893 183818 597261 675776 718872 173918 289686 149496 > 16¹⁰⁹ [i]