MinT - Best Known (53, 53+67, s)-Nets in Base 16

Best Known (53, 53+67, s)-Nets in Base 16

(53, 53+67, 243)-Net over F₁₆ — Constructive and digital

Digital (53, 120, 243)-net over F₁₆, using

t-expansion [i] based on digital (48, 120, 243)-net over F₁₆, using
- net from sequence [i] based on digital (48, 242)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 48 and N(F) ≥ 243, using
    - a function field by GarcÃa and Quoos [i]

(53, 53+67, 255)-Net over F₁₆ — Digital

Digital (53, 120, 255)-net over F₁₆, using

t-expansion [i] based on digital (50, 120, 255)-net over F₁₆, using
- net from sequence [i] based on digital (50, 254)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 50 and N(F) ≥ 255, using
    - a curve from the manYPoints database [i]

(53, 53+67, 257)-Net in Base 16

(53, 120, 257)-net in base 16, using

5 times m-reduction [i] based on (53, 125, 257)-net in base 16, using
- base change [i] based on digital (28, 100, 257)-net over F₃₂, using
  - net from sequence [i] based on digital (28, 256)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 28 and N(F) ≥ 257, using
      - a curve from the manYPoints database [i]

(53, 53+67, 19274)-Net in Base 16 — Upper bound on s

There is no (53, 120, 19275)-net in base 16, because

1 times m-reduction [i] would yield (53, 119, 19275)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 195282 561018 618509 124795 661040 876380 796914 064440 324275 917857 337982 701851 324772 808283 134589 572141 994444 765880 558723 172831 017670 809132 249791 519876 > 16¹¹⁹ [i]