MinT - Best Known (75−40, 75, s)-Nets in Base 16

Best Known (75−40, 75, s)-Nets in Base 16

(75−40, 75, 130)-Net over F₁₆ — Constructive and digital

Digital (35, 75, 130)-net over F₁₆, using

6 times m-reduction [i] based on digital (35, 81, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 29, 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, 52, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(75−40, 75, 177)-Net in Base 16 — Constructive

(35, 75, 177)-net in base 16, using

9 times m-reduction [i] based on (35, 84, 177)-net in base 16, using
- base change [i] based on digital (7, 56, 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]

(75−40, 75, 207)-Net over F₁₆ — Digital

Digital (35, 75, 207)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−40, 75, 225)-Net in Base 16

(35, 75, 225)-net in base 16, using

base change [i] based on digital (10, 50, 225)-net over F₆₄, using
- net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(75−40, 75, 18130)-Net in Base 16 — Upper bound on s

There is no (35, 75, 18131)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 038558 718850 698140 366346 689267 066499 532042 971855 426625 262597 463988 495510 612439 134589 400176 > 16⁷⁵ [i]