MinT - Best Known (27, 52, s)-Nets in Base 16

Best Known (27, 52, s)-Nets in Base 16

(27, 52, 516)-Net over F₁₆ — Constructive and digital

Digital (27, 52, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 26, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(27, 52, 578)-Net over F₁₆ — Digital

Digital (27, 52, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 26, 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]

(27, 52, 46208)-Net in Base 16 — Upper bound on s

There is no (27, 52, 46209)-net in base 16, because

1 times m-reduction [i] would yield (27, 51, 46209)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 25 715053 789991 701834 921129 408705 993824 795457 284365 454705 966496 > 16⁵¹ [i]