MinT - Best Known (39, 88, s)-Nets in Base 16

Best Known (39, 88, s)-Nets in Base 16

(39, 88, 130)-Net over F₁₆ — Constructive and digital

Digital (39, 88, 130)-net over F₁₆, using

5 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, 88, 177)-Net in Base 16 — Constructive

(39, 88, 177)-net in base 16, using

8 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, 88, 208)-Net over F₁₆ — Digital

Digital (39, 88, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 88, 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, 88, 209)-Net in Base 16

(39, 88, 209)-net in base 16, using

2 times m-reduction [i] based on (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, 88, 15129)-Net in Base 16 — Upper bound on s

There is no (39, 88, 15130)-net in base 16, because

1 times m-reduction [i] would yield (39, 87, 15130)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 573 637498 994538 643652 401638 783339 877053 868244 917495 862981 405572 769868 702241 166256 409599 436952 133155 425801 > 16⁸⁷ [i]