MinT - Best Known (31, 57, s)-Nets in Base 7

Best Known (31, 57, s)-Nets in Base 7

(31, 57, 104)-Net over F₇ — Constructive and digital

Digital (31, 57, 104)-net over F₇, using

1 times m-reduction [i] based on digital (31, 58, 104)-net over F₇, using
- trace code for nets [i] based on digital (2, 29, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(31, 57, 156)-Net over F₇ — Digital

Digital (31, 57, 156)-net over F₇, using

1 times m-reduction [i] based on digital (31, 58, 156)-net over F₇, using
- trace code for nets [i] based on digital (2, 29, 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]

(31, 57, 4785)-Net in Base 7 — Upper bound on s

There is no (31, 57, 4786)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1 482351 376696 138219 078837 803173 884501 057054 446229 > 7⁵⁷ [i]