MinT - Best Known (69−67, 69, s)-Nets in Base 7

Best Known (69−67, 69, s)-Nets in Base 7

(69−67, 69, 10)-Net over F₇ — Constructive and digital

Digital (2, 69, 10)-net over F₇, using

net from sequence [i] based on digital (2, 9)-sequence over F₇, using
- Niederreiter sequence [i]

(69−67, 69, 16)-Net over F₇ — Digital

Digital (2, 69, 16)-net over F₇, using

net from sequence [i] based on digital (2, 15)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 2 and N(F) ≥ 16, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(69−67, 69, 20)-Net in Base 7 — Upper bound on s

There is no (2, 69, 21)-net in base 7, because

45 times m-reduction [i] would yield (2, 24, 21)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(7²⁴, 21, S₇, 2, 22), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 185642 213207 768855 554569 / 961 > 7²⁴ [i]