MinT - Best Known (53−25, 53, s)-Nets in Base 7

Best Known (53−25, 53, s)-Nets in Base 7

(53−25, 53, 102)-Net over F₇ — Constructive and digital

Digital (28, 53, 102)-net over F₇, using

1 times m-reduction [i] based on digital (28, 54, 102)-net over F₇, using
- trace code for nets [i] based on digital (1, 27, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(53−25, 53, 128)-Net over F₇ — Digital

Digital (28, 53, 128)-net over F₇, using

1 times m-reduction [i] based on digital (28, 54, 128)-net over F₇, using
- trace code for nets [i] based on digital (1, 27, 64)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 63)-sequence over F₄₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(53−25, 53, 4041)-Net in Base 7 — Upper bound on s

There is no (28, 53, 4042)-net in base 7, because

1 times m-reduction [i] would yield (28, 52, 4042)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 88 366756 088033 500099 607450 797849 107219 570457 > 7⁵² [i]