MinT - Best Known (20−9, 20, s)-Nets in Base 4

Best Known (20−9, 20, s)-Nets in Base 4

(20−9, 20, 48)-Net over F₄ — Constructive and digital

Digital (11, 20, 48)-net over F₄, using

trace code for nets [i] based on digital (1, 10, 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]

(20−9, 20, 50)-Net over F₄ — Digital

Digital (11, 20, 50)-net over F₄, using

trace code for nets [i] based on digital (1, 10, 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]

(20−9, 20, 531)-Net in Base 4 — Upper bound on s

There is no (11, 20, 532)-net in base 4, because

1 times m-reduction [i] would yield (11, 19, 532)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 276138 961204 > 4¹⁹ [i]