MinT - Best Known (9, 9+7, s)-Nets in Base 16

Best Known (9, 9+7, s)-Nets in Base 16

(9, 9+7, 516)-Net over F₁₆ — Constructive and digital

Digital (9, 16, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 8, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(9, 9+7, 578)-Net over F₁₆ — Digital

Digital (9, 16, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 8, 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]

(9, 9+7, 127024)-Net in Base 16 — Upper bound on s

There is no (9, 16, 127025)-net in base 16, because

1 times m-reduction [i] would yield (9, 15, 127025)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 152928 687918 107376 > 16¹⁵ [i]