MinT - Best Known (54−25, 54, s)-Nets in Base 7

Best Known (54−25, 54, s)-Nets in Base 7

(54−25, 54, 104)-Net over F₇ — Constructive and digital

Digital (29, 54, 104)-net over F₇, using

trace code for nets [i] based on digital (2, 27, 52)-net over F₄₉, using
- net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
  - Niederreiter sequence [i]

(54−25, 54, 156)-Net over F₇ — Digital

Digital (29, 54, 156)-net over F₇, using

trace code for nets [i] based on digital (2, 27, 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]

(54−25, 54, 4753)-Net in Base 7 — Upper bound on s

There is no (29, 54, 4754)-net in base 7, because

1 times m-reduction [i] would yield (29, 53, 4754)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 617 162014 066827 621522 554738 822032 271874 639001 > 7⁵³ [i]