MinT - Best Known (16, 29, s)-Nets in Base 4

Best Known (16, 29, s)-Nets in Base 4

(16, 29, 48)-Net over F₄ — Constructive and digital

Digital (16, 29, 48)-net over F₄, using

1 times m-reduction [i] based on digital (16, 30, 48)-net over F₄, using
- trace code for nets [i] based on digital (1, 15, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]

(16, 29, 50)-Net over F₄ — Digital

Digital (16, 29, 50)-net over F₄, using

1 times m-reduction [i] based on digital (16, 30, 50)-net over F₄, using
- trace code for nets [i] based on digital (1, 15, 25)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 24)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 25, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(16, 29, 639)-Net in Base 4 — Upper bound on s

There is no (16, 29, 640)-net in base 4, because

1 times m-reduction [i] would yield (16, 28, 640)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 72557 403726 165985 > 4²⁸ [i]