MinT - Best Known (77−37, 77, s)-Nets in Base 16

Best Known (77−37, 77, s)-Nets in Base 16

(77−37, 77, 516)-Net over F₁₆ — Constructive and digital

Digital (40, 77, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (40, 78, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 39, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(77−37, 77, 578)-Net over F₁₆ — Digital

Digital (40, 77, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (40, 78, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 39, 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]

(77−37, 77, 61098)-Net in Base 16 — Upper bound on s

There is no (40, 77, 61099)-net in base 16, because

1 times m-reduction [i] would yield (40, 76, 61099)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 32 595406 283907 432573 117970 456477 650354 308408 200096 352318 712295 921186 023773 778022 291037 991106 > 16⁷⁶ [i]