MinT - Best Known (66−31, 66, s)-Nets in Base 7

Best Known (66−31, 66, s)-Nets in Base 7

(66−31, 66, 104)-Net over F₇ — Constructive and digital

Digital (35, 66, 104)-net over F₇, using

trace code for nets [i] based on digital (2, 33, 52)-net over F₄₉, using
- net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
  - Niederreiter sequence [i]

(66−31, 66, 156)-Net over F₇ — Digital

Digital (35, 66, 156)-net over F₇, using

trace code for nets [i] based on digital (2, 33, 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]

(66−31, 66, 4907)-Net in Base 7 — Upper bound on s

There is no (35, 66, 4908)-net in base 7, because

1 times m-reduction [i] would yield (35, 65, 4908)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 8 549223 394025 329760 330043 866528 619089 738647 409989 548385 > 7⁶⁵ [i]