MinT - Best Known (51, 51+69, s)-Nets in Base 16

Best Known (51, 51+69, s)-Nets in Base 16

(51, 51+69, 243)-Net over F₁₆ — Constructive and digital

Digital (51, 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]

(51, 51+69, 255)-Net over F₁₆ — Digital

Digital (51, 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]

(51, 51+69, 14765)-Net in Base 16 — Upper bound on s

There is no (51, 120, 14766)-net in base 16, because

1 times m-reduction [i] would yield (51, 119, 14766)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 195265 000165 342626 228640 806416 120946 199061 853720 137679 924071 447572 317827 299261 796902 082755 259185 777741 444555 680194 662358 353320 109609 038855 933886 > 16¹¹⁹ [i]