MinT - Best Known (57−27, 57, s)-Nets in Base 16

Best Known (57−27, 57, s)-Nets in Base 16

(57−27, 57, 516)-Net over F₁₆ — Constructive and digital

Digital (30, 57, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (30, 58, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 29, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(57−27, 57, 578)-Net over F₁₆ — Digital

Digital (30, 57, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (30, 58, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 29, 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]

(57−27, 57, 58108)-Net in Base 16 — Upper bound on s

There is no (30, 57, 58109)-net in base 16, because

1 times m-reduction [i] would yield (30, 56, 58109)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 26 961085 493676 750505 003400 392311 067018 206970 089703 347885 528735 372256 > 16⁵⁶ [i]