MinT - Best Known (81−45, 81, s)-Nets in Base 16

Best Known (81−45, 81, s)-Nets in Base 16

(81−45, 81, 130)-Net over F₁₆ — Constructive and digital

Digital (36, 81, 130)-net over F₁₆, using

3 times m-reduction [i] based on digital (36, 84, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 30, 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]
  2. digital (6, 54, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(81−45, 81, 177)-Net in Base 16 — Constructive

(36, 81, 177)-net in base 16, using

6 times m-reduction [i] based on (36, 87, 177)-net in base 16, using
- base change [i] based on digital (7, 58, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(81−45, 81, 193)-Net over F₁₆ — Digital

Digital (36, 81, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 81, 193)-net over F₁₆, using
- net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
    - a curve from the manYPoints database [i]

(81−45, 81, 209)-Net in Base 16

(36, 81, 209)-net in base 16, using

base change [i] based on digital (9, 54, 209)-net over F₆₄, using
- net from sequence [i] based on digital (9, 208)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 9 and N(F) ≥ 209, using
    - a curve from the manYPoints database [i]

(81−45, 81, 14421)-Net in Base 16 — Upper bound on s

There is no (36, 81, 14422)-net in base 16, because

1 times m-reduction [i] would yield (36, 80, 14422)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 136607 690707 077031 102456 612180 030668 285095 491170 111343 962143 934541 646605 402791 059445 224432 423936 > 16⁸⁰ [i]