MinT - Best Known (69−47, 69, s)-Nets in Base 16

Best Known (69−47, 69, s)-Nets in Base 16

(69−47, 69, 65)-Net over F₁₆ — Constructive and digital

Digital (22, 69, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 69, 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]

(69−47, 69, 98)-Net in Base 16 — Constructive

(22, 69, 98)-net in base 16, using

6 times m-reduction [i] based on (22, 75, 98)-net in base 16, using
- base change [i] based on digital (7, 60, 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]

(69−47, 69, 129)-Net over F₁₆ — Digital

Digital (22, 69, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 69, 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]

(69−47, 69, 2269)-Net in Base 16 — Upper bound on s

There is no (22, 69, 2270)-net in base 16, because

1 times m-reduction [i] would yield (22, 68, 2270)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7605 397750 109954 274866 564304 103477 030931 057901 845927 136188 549310 608541 513284 794276 > 16⁶⁸ [i]