MinT - Best Known (85−61, 85, s)-Nets in Base 16

Best Known (85−61, 85, s)-Nets in Base 16

(85−61, 85, 65)-Net over F₁₆ — Constructive and digital

Digital (24, 85, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 85, 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]

(85−61, 85, 98)-Net in Base 16 — Constructive

(24, 85, 98)-net in base 16, using

base change [i] based on digital (7, 68, 98)-net over F₃₂, using
- net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
    - a function field by SÃ©mirat [i]

(85−61, 85, 129)-Net over F₁₆ — Digital

Digital (24, 85, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 85, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(85−61, 85, 1872)-Net in Base 16 — Upper bound on s

There is no (24, 85, 1873)-net in base 16, because

1 times m-reduction [i] would yield (24, 84, 1873)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 141005 359903 854459 579727 413755 369821 298352 408272 637612 391226 355847 500987 721920 387120 505777 283372 346976 > 16⁸⁴ [i]