MinT - Best Known (52−25, 52, s)-Nets in Base 8

Best Known (52−25, 52, s)-Nets in Base 8

(52−25, 52, 160)-Net over F₈ — Constructive and digital

Digital (27, 52, 160)-net over F₈, using

trace code for nets [i] based on digital (1, 26, 80)-net over F₆₄, using
- net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
    - a function field by SÃ©mirat [i]

(52−25, 52, 162)-Net over F₈ — Digital

Digital (27, 52, 162)-net over F₈, using

trace code for nets [i] based on digital (1, 26, 81)-net over F₆₄, using
- net from sequence [i] based on digital (1, 80)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 81, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(52−25, 52, 5197)-Net in Base 8 — Upper bound on s

There is no (27, 52, 5198)-net in base 8, because

1 times m-reduction [i] would yield (27, 51, 5198)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11431 406157 941259 704913 304848 646987 919074 908597 > 8⁵¹ [i]