MinT - Best Known (76−37, 76, s)-Nets in Base 8

Best Known (76−37, 76, s)-Nets in Base 8

(76−37, 76, 160)-Net over F₈ — Constructive and digital

Digital (39, 76, 160)-net over F₈, using

trace code for nets [i] based on digital (1, 38, 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]

(76−37, 76, 162)-Net over F₈ — Digital

Digital (39, 76, 162)-net over F₈, using

trace code for nets [i] based on digital (1, 38, 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]

(76−37, 76, 6239)-Net in Base 8 — Upper bound on s

There is no (39, 76, 6240)-net in base 8, because

1 times m-reduction [i] would yield (39, 75, 6240)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 54 025192 350373 823342 307724 365993 130125 919719 951423 591500 078455 649191 > 8⁷⁵ [i]