MinT - Best Known (81−39, 81, s)-Nets in Base 16

Best Known (81−39, 81, s)-Nets in Base 16

(81−39, 81, 516)-Net over F₁₆ — Constructive and digital

Digital (42, 81, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (42, 82, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 41, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(81−39, 81, 578)-Net over F₁₆ — Digital

Digital (42, 81, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (42, 82, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 41, 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]

(81−39, 81, 62091)-Net in Base 16 — Upper bound on s

There is no (42, 81, 62092)-net in base 16, because

1 times m-reduction [i] would yield (42, 80, 62092)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 136391 424477 494246 442661 415697 449265 420676 077040 055219 518231 257776 069938 677191 199224 373650 391896 > 16⁸⁰ [i]