MinT - Best Known (69−37, 69, s)-Nets in Base 16

Best Known (69−37, 69, s)-Nets in Base 16

(69−37, 69, 130)-Net over F₁₆ — Constructive and digital

Digital (32, 69, 130)-net over F₁₆, using

3 times m-reduction [i] based on digital (32, 72, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 26, 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, 46, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(69−37, 69, 177)-Net in Base 16 — Constructive

(32, 69, 177)-net in base 16, using

6 times m-reduction [i] based on (32, 75, 177)-net in base 16, using
- base change [i] based on digital (7, 50, 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]

(69−37, 69, 189)-Net over F₁₆ — Digital

Digital (32, 69, 189)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−37, 69, 209)-Net in Base 16

(32, 69, 209)-net in base 16, using

base change [i] based on digital (9, 46, 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]

(69−37, 69, 17811)-Net in Base 16 — Upper bound on s

There is no (32, 69, 17812)-net in base 16, because

1 times m-reduction [i] would yield (32, 68, 17812)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7592 509798 135320 856886 211618 744393 585625 403078 303354 011105 775449 828031 522708 062891 > 16⁶⁸ [i]