MinT - Best Known (70−33, 70, s)-Nets in Base 16

Best Known (70−33, 70, s)-Nets in Base 16

(70−33, 70, 518)-Net over F₁₆ — Constructive and digital

Digital (37, 70, 518)-net over F₁₆, using

trace code for nets [i] based on digital (2, 35, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(70−33, 70, 642)-Net over F₁₆ — Digital

Digital (37, 70, 642)-net over F₁₆, using

trace code for nets [i] based on digital (2, 35, 321)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(70−33, 70, 70658)-Net in Base 16 — Upper bound on s

There is no (37, 70, 70659)-net in base 16, because

1 times m-reduction [i] would yield (37, 69, 70659)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 121440 269120 344377 687003 820613 275479 929480 265666 959236 629023 492791 531773 980595 257536 > 16⁶⁹ [i]