MinT - Best Known (64−31, 64, s)-Nets in Base 16

Best Known (64−31, 64, s)-Nets in Base 16

(64−31, 64, 516)-Net over F₁₆ — Constructive and digital

Digital (33, 64, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 32, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(64−31, 64, 578)-Net over F₁₆ — Digital

Digital (33, 64, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 32, 289)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(64−31, 64, 48854)-Net in Base 16 — Upper bound on s

There is no (33, 64, 48855)-net in base 16, because

1 times m-reduction [i] would yield (33, 63, 48855)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7237 209437 384509 213570 425485 132410 490783 305939 873297 999274 941635 301133 787376 > 16⁶³ [i]