MinT - Best Known (70−33, 70, s)-Nets in Base 7

Best Known (70−33, 70, s)-Nets in Base 7

(70−33, 70, 104)-Net over F₇ — Constructive and digital

Digital (37, 70, 104)-net over F₇, using

trace code for nets [i] based on digital (2, 35, 52)-net over F₄₉, using
- net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
  - Niederreiter sequence [i]

(70−33, 70, 156)-Net over F₇ — Digital

Digital (37, 70, 156)-net over F₇, using

trace code for nets [i] based on digital (2, 35, 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]

(70−33, 70, 4988)-Net in Base 7 — Upper bound on s

There is no (37, 70, 4989)-net in base 7, because

1 times m-reduction [i] would yield (37, 69, 4989)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 20518 841315 274014 793090 850996 574225 368323 602908 989430 397185 > 7⁶⁹ [i]