MinT - Best Known (41−27, 41, s)-Nets in Base 49

Best Known (41−27, 41, s)-Nets in Base 49

(41−27, 41, 101)-Net over F₄₉ — Constructive and digital

Digital (14, 41, 101)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 13, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
      - the rational function field F₄₉(x) [i]
    - Niederreiter sequence [i]
2. digital (1, 28, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(41−27, 41, 114)-Net over F₄₉ — Digital

Digital (14, 41, 114)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 13, 50)-net over F₄₉, using
  - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
      - the rational function field F₄₉(x) [i]
    - Niederreiter sequence [i]
2. digital (1, 28, 64)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 63)-sequence over F₄₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(41−27, 41, 18733)-Net in Base 49 — Upper bound on s

There is no (14, 41, 18734)-net in base 49, because

1 times m-reduction [i] would yield (14, 40, 18734)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 40 545424 268891 864408 793852 753025 362265 600961 533058 707098 118920 842785 > 49⁴⁰ [i]