MinT - Best Known (18, 18+15, s)-Nets in Base 16

Best Known (18, 18+15, s)-Nets in Base 16

(18, 18+15, 516)-Net over F₁₆ — Constructive and digital

Digital (18, 33, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (18, 34, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 17, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(18, 18+15, 578)-Net over F₁₆ — Digital

Digital (18, 33, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (18, 34, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 17, 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]

(18, 18+15, 72003)-Net in Base 16 — Upper bound on s

There is no (18, 33, 72004)-net in base 16, because

1 times m-reduction [i] would yield (18, 32, 72004)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 340 291420 095942 040405 598637 963982 973296 > 16³² [i]