MinT - Best Known (65−31, 65, s)-Nets in Base 16

Best Known (65−31, 65, s)-Nets in Base 16

(65−31, 65, 516)-Net over F₁₆ — Constructive and digital

Digital (34, 65, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (34, 66, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 33, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(65−31, 65, 578)-Net over F₁₆ — Digital

Digital (34, 65, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (34, 66, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 33, 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]

(65−31, 65, 58775)-Net in Base 16 — Upper bound on s

There is no (34, 65, 58776)-net in base 16, because

1 times m-reduction [i] would yield (34, 64, 58776)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 115812 720205 258939 345426 295699 217667 577590 930510 459603 445338 141555 418660 230726 > 16⁶⁴ [i]