MinT - Best Known (61, 61+57, s)-Nets in Base 16

Best Known (61, 61+57, s)-Nets in Base 16

(61, 61+57, 518)-Net over F₁₆ — Constructive and digital

Digital (61, 118, 518)-net over F₁₆, using

trace code for nets [i] based on digital (2, 59, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(61, 61+57, 642)-Net over F₁₆ — Digital

Digital (61, 118, 642)-net over F₁₆, using

trace code for nets [i] based on digital (2, 59, 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]

(61, 61+57, 80971)-Net in Base 16 — Upper bound on s

There is no (61, 118, 80972)-net in base 16, because

1 times m-reduction [i] would yield (61, 117, 80972)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 762 251177 041066 171601 816588 642173 027466 883420 659387 551365 617623 116022 827451 354328 611796 739708 929884 329722 089883 597309 875305 304037 328246 360016 > 16¹¹⁷ [i]