MinT - Best Known (49−23, 49, s)-Nets in Base 8

Best Known (49−23, 49, s)-Nets in Base 8

(49−23, 49, 160)-Net over F₈ — Constructive and digital

Digital (26, 49, 160)-net over F₈, using

1 times m-reduction [i] based on digital (26, 50, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 25, 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]

(49−23, 49, 162)-Net over F₈ — Digital

Digital (26, 49, 162)-net over F₈, using

1 times m-reduction [i] based on digital (26, 50, 162)-net over F₈, using
- trace code for nets [i] based on digital (1, 25, 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]

(49−23, 49, 6112)-Net in Base 8 — Upper bound on s

There is no (26, 49, 6113)-net in base 8, because

1 times m-reduction [i] would yield (26, 48, 6113)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 22 328113 485374 184216 585726 713408 347009 260384 > 8⁴⁸ [i]