MinT - Best Known (53−25, 53, s)-Nets in Base 8

Best Known (53−25, 53, s)-Nets in Base 8

(53−25, 53, 160)-Net over F₈ — Constructive and digital

Digital (28, 53, 160)-net over F₈, using

1 times m-reduction [i] based on digital (28, 54, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 27, 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]

(53−25, 53, 162)-Net over F₈ — Digital

Digital (28, 53, 162)-net over F₈, using

1 times m-reduction [i] based on digital (28, 54, 162)-net over F₈, using
- trace code for nets [i] based on digital (1, 27, 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]

(53−25, 53, 6182)-Net in Base 8 — Upper bound on s

There is no (28, 53, 6183)-net in base 8, because

1 times m-reduction [i] would yield (28, 52, 6183)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 91475 636375 977337 852604 888342 050045 020972 513612 > 8⁵² [i]