MinT - Best Known (37, 72, s)-Nets in Base 16

Best Known (37, 72, s)-Nets in Base 16

(37, 72, 516)-Net over F₁₆ — Constructive and digital

Digital (37, 72, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 36, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(37, 72, 578)-Net over F₁₆ — Digital

Digital (37, 72, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 36, 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]

(37, 72, 51138)-Net in Base 16 — Upper bound on s

There is no (37, 72, 51139)-net in base 16, because

1 times m-reduction [i] would yield (37, 71, 51139)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 31 087745 958064 599000 061638 204059 639657 665665 496578 705905 505999 964872 197713 838796 184696 > 16⁷¹ [i]