MinT - Best Known (27, 27+91, s)-Nets in Base 9

Best Known (27, 27+91, s)-Nets in Base 9

(27, 27+91, 78)-Net over F₉ — Constructive and digital

Digital (27, 118, 78)-net over F₉, using

t-expansion [i] based on digital (22, 118, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(27, 27+91, 110)-Net over F₉ — Digital

Digital (27, 118, 110)-net over F₉, using

t-expansion [i] based on digital (26, 118, 110)-net over F₉, using
- net from sequence [i] based on digital (26, 109)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 26 and N(F) ≥ 110, using
    - a curve from the manYPoints database [i]

(27, 27+91, 639)-Net in Base 9 — Upper bound on s

There is no (27, 118, 640)-net in base 9, because

1 times m-reduction [i] would yield (27, 117, 640)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4524 318759 880785 171548 205731 538830 583020 084698 506752 954587 593791 510186 127769 055817 945144 882918 677169 432955 511809 > 9¹¹⁷ [i]