MinT - Best Known (59−27, 59, s)-Nets in Base 7

Best Known (59−27, 59, s)-Nets in Base 7

(59−27, 59, 104)-Net over F₇ — Constructive and digital

Digital (32, 59, 104)-net over F₇, using

1 times m-reduction [i] based on digital (32, 60, 104)-net over F₇, using
- trace code for nets [i] based on digital (2, 30, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(59−27, 59, 156)-Net over F₇ — Digital

Digital (32, 59, 156)-net over F₇, using

1 times m-reduction [i] based on digital (32, 60, 156)-net over F₇, using
- trace code for nets [i] based on digital (2, 30, 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]

(59−27, 59, 5559)-Net in Base 7 — Upper bound on s

There is no (32, 59, 5560)-net in base 7, because

1 times m-reduction [i] would yield (32, 58, 5560)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 10 373871 022135 946785 495283 525183 717631 397402 168273 > 7⁵⁸ [i]