MinT - Best Known (85−41, 85, s)-Nets in Base 16

Best Known (85−41, 85, s)-Nets in Base 16

(85−41, 85, 516)-Net over F₁₆ — Constructive and digital

Digital (44, 85, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (44, 86, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 43, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(85−41, 85, 578)-Net over F₁₆ — Digital

Digital (44, 85, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (44, 86, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 43, 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]

(85−41, 85, 63160)-Net in Base 16 — Upper bound on s

There is no (44, 85, 63161)-net in base 16, because

1 times m-reduction [i] would yield (44, 84, 63161)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 140013 565954 658173 585731 161556 215175 928976 642435 784635 795616 251935 248537 954900 347357 986330 018177 007926 > 16⁸⁴ [i]