MinT - Best Known (82−54, 82, s)-Nets in Base 16

Best Known (82−54, 82, s)-Nets in Base 16

(82−54, 82, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 82, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 82, 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]

(82−54, 82, 120)-Net in Base 16 — Constructive

(28, 82, 120)-net in base 16, using

3 times m-reduction [i] based on (28, 85, 120)-net in base 16, using
- base change [i] based on digital (11, 68, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(82−54, 82, 156)-Net over F₁₆ — Digital

Digital (28, 82, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 82, 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]

(82−54, 82, 3290)-Net in Base 16 — Upper bound on s

There is no (28, 82, 3291)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 546 876780 322133 885425 089205 886181 752325 249916 407698 059525 076709 971565 133887 599985 706446 542860 033456 > 16⁸² [i]