MinT - Best Known (50−23, 50, s)-Nets in Base 7

Best Known (50−23, 50, s)-Nets in Base 7

(50−23, 50, 104)-Net over F₇ — Constructive and digital

Digital (27, 50, 104)-net over F₇, using

trace code for nets [i] based on digital (2, 25, 52)-net over F₄₉, using
- net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
  - Niederreiter sequence [i]

(50−23, 50, 156)-Net over F₇ — Digital

Digital (27, 50, 156)-net over F₇, using

trace code for nets [i] based on digital (2, 25, 78)-net over F₄₉, using
- net from sequence [i] based on digital (2, 77)-sequence over F₄₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 2 and N(F) ≥ 78, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(50−23, 50, 4750)-Net in Base 7 — Upper bound on s

There is no (27, 50, 4751)-net in base 7, because

1 times m-reduction [i] would yield (27, 49, 4751)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 257043 246124 405636 769504 175746 960262 254143 > 7⁴⁹ [i]