MinT - Best Known (109−53, 109, s)-Nets in Base 16

Best Known (109−53, 109, s)-Nets in Base 16

(109−53, 109, 516)-Net over F₁₆ — Constructive and digital

Digital (56, 109, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (56, 110, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 55, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(109−53, 109, 578)-Net over F₁₆ — Digital

Digital (56, 109, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (56, 110, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 55, 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]

(109−53, 109, 70606)-Net in Base 16 — Upper bound on s

There is no (56, 109, 70607)-net in base 16, because

1 times m-reduction [i] would yield (56, 108, 70607)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11093 570212 614137 093666 619754 791103 029321 255007 578775 482402 370039 029833 353149 092703 517018 816499 831045 278251 700263 242500 355580 137356 > 16¹⁰⁸ [i]