MinT - Best Known (53, 53+69, s)-Nets in Base 16

Best Known (53, 53+69, s)-Nets in Base 16

(53, 53+69, 243)-Net over F₁₆ — Constructive and digital

Digital (53, 122, 243)-net over F₁₆, using

t-expansion [i] based on digital (48, 122, 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+69, 255)-Net over F₁₆ — Digital

Digital (53, 122, 255)-net over F₁₆, using

t-expansion [i] based on digital (50, 122, 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+69, 257)-Net in Base 16

(53, 122, 257)-net in base 16, using

3 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+69, 17384)-Net in Base 16 — Upper bound on s

There is no (53, 122, 17385)-net in base 16, because

1 times m-reduction [i] would yield (53, 121, 17385)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 49 980947 392440 334047 243477 824056 975542 997909 173630 954599 712371 294272 632604 220316 074620 972450 766188 784041 732258 589344 394365 390854 384479 322598 997476 > 16¹²¹ [i]