MinT - Best Known (45, 45+42, s)-Nets in Base 16

Best Known (45, 45+42, s)-Nets in Base 16

(45, 45+42, 516)-Net over F₁₆ — Constructive and digital

Digital (45, 87, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (45, 88, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 44, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(45, 45+42, 578)-Net over F₁₆ — Digital

Digital (45, 87, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (45, 88, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 44, 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]

(45, 45+42, 56339)-Net in Base 16 — Upper bound on s

There is no (45, 87, 56340)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 573 550082 613920 664431 011746 727076 035272 171420 263263 114269 423993 392934 629658 588168 857696 295277 176263 838976 > 16⁸⁷ [i]