MinT - Best Known (51−23, 51, s)-Nets in Base 7

Best Known (51−23, 51, s)-Nets in Base 7

(51−23, 51, 104)-Net over F₇ — Constructive and digital

Digital (28, 51, 104)-net over F₇, using

1 times m-reduction [i] based on digital (28, 52, 104)-net over F₇, using
- trace code for nets [i] based on digital (2, 26, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(51−23, 51, 156)-Net over F₇ — Digital

Digital (28, 51, 156)-net over F₇, using

1 times m-reduction [i] based on digital (28, 52, 156)-net over F₇, using
- trace code for nets [i] based on digital (2, 26, 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]

(51−23, 51, 5671)-Net in Base 7 — Upper bound on s

There is no (28, 51, 5672)-net in base 7, because

1 times m-reduction [i] would yield (28, 50, 5672)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1 800346 355813 790713 463020 677160 610882 289153 > 7⁵⁰ [i]