MinT - Best Known (65−63, 65, s)-Nets in Base 8

Best Known (65−63, 65, s)-Nets in Base 8

(65−63, 65, 17)-Net over F₈ — Constructive and digital

Digital (2, 65, 17)-net over F₈, using

net from sequence [i] based on digital (2, 16)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
  - a function field by SÃ©mirat [i]

(65−63, 65, 18)-Net over F₈ — Digital

Digital (2, 65, 18)-net over F₈, using

net from sequence [i] based on digital (2, 17)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 18, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(65−63, 65, 23)-Net in Base 8 — Upper bound on s

There is no (2, 65, 24)-net in base 8, because

38 times m-reduction [i] would yield (2, 27, 24)-net in base 8, but
- extracting embedded OOA [i] would yield OOA(8²⁷, 24, S₈, 2, 25), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 1329 818401 576092 092176 793600 / 513 > 8²⁷ [i]