MinT - Best Known (39, 39+51, s)-Nets in Base 16

Best Known (39, 39+51, s)-Nets in Base 16

(39, 39+51, 130)-Net over F₁₆ — Constructive and digital

Digital (39, 90, 130)-net over F₁₆, using

3 times m-reduction [i] based on digital (39, 93, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 33, 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, 60, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(39, 39+51, 177)-Net in Base 16 — Constructive

(39, 90, 177)-net in base 16, using

6 times m-reduction [i] based on (39, 96, 177)-net in base 16, using
- base change [i] based on digital (7, 64, 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]

(39, 39+51, 208)-Net over F₁₆ — Digital

Digital (39, 90, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 90, 208)-net over F₁₆, using
- net from sequence [i] based on digital (37, 207)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 37 and N(F) ≥ 208, using
    - Drinfeld modules of rank 1 [i]

(39, 39+51, 209)-Net in Base 16

(39, 90, 209)-net in base 16, using

base change [i] based on digital (9, 60, 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]

(39, 39+51, 13114)-Net in Base 16 — Upper bound on s

There is no (39, 90, 13115)-net in base 16, because

1 times m-reduction [i] would yield (39, 89, 13115)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146942 817565 020151 370901 218559 540462 205558 195778 555237 189963 623182 584875 184344 180032 856733 742755 527270 580626 > 16⁸⁹ [i]