MinT - Best Known (67−31, 67, s)-Nets in Base 7

Best Known (67−31, 67, s)-Nets in Base 7

(67−31, 67, 104)-Net over F₇ — Constructive and digital

Digital (36, 67, 104)-net over F₇, using

1 times m-reduction [i] based on digital (36, 68, 104)-net over F₇, using
- trace code for nets [i] based on digital (2, 34, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(67−31, 67, 156)-Net over F₇ — Digital

Digital (36, 67, 156)-net over F₇, using

1 times m-reduction [i] based on digital (36, 68, 156)-net over F₇, using
- trace code for nets [i] based on digital (2, 34, 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]

(67−31, 67, 5588)-Net in Base 7 — Upper bound on s

There is no (36, 67, 5589)-net in base 7, because

1 times m-reduction [i] would yield (36, 66, 5589)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 59 818869 874225 120675 566708 095604 667111 133136 355893 664663 > 7⁶⁶ [i]