MinT - Best Known (89−43, 89, s)-Nets in Base 16

Best Known (89−43, 89, s)-Nets in Base 16

(89−43, 89, 516)-Net over F₁₆ — Constructive and digital

Digital (46, 89, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (46, 90, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 45, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(89−43, 89, 578)-Net over F₁₆ — Digital

Digital (46, 89, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (46, 90, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 45, 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]

(89−43, 89, 64292)-Net in Base 16 — Upper bound on s

There is no (46, 89, 64293)-net in base 16, because

1 times m-reduction [i] would yield (46, 88, 64293)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9175 441116 977630 696844 434696 753742 142508 066998 898260 190528 220388 090439 701186 020974 208676 373229 059317 463296 > 16⁸⁸ [i]