MinT - Best Known (72−31, 72, s)-Nets in Base 8

Best Known (72−31, 72, s)-Nets in Base 8

(72−31, 72, 256)-Net over F₈ — Constructive and digital

Digital (41, 72, 256)-net over F₈, using

trace code for nets [i] based on digital (5, 36, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(72−31, 72, 266)-Net over F₈ — Digital

Digital (41, 72, 266)-net over F₈, using

trace code for nets [i] based on digital (5, 36, 133)-net over F₆₄, using
- net from sequence [i] based on digital (5, 132)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 133, using
    - a curve from the manYPoints database [i]

(72−31, 72, 17260)-Net in Base 8 — Upper bound on s

There is no (41, 72, 17261)-net in base 8, because

1 times m-reduction [i] would yield (41, 71, 17261)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 13164 651336 056115 785809 009496 373835 527033 359171 169207 062767 567136 > 8⁷¹ [i]